Proof. Summing over the implicit geometric progression, we find that when $\alpha \notin \mathbb Z$ , then for integral values of N one has

(2.1) $$ \begin{align} \bigg| \sum_{1\leqslant n\leqslant N}e(n\alpha)\bigg| =\Big| \frac{\sin(N\pi\alpha)}{\sin(\pi \alpha)}\Big| \ll \|\alpha\|^{-1}. \end{align} $$

Thus

$$\begin{align*}\int_0^1\bigg| \sum_{1\leqslant n\leqslant N}e(n\alpha)\bigg|{\,\mathrm{d}}\alpha \ll 1+\int_{1/N}^{1/2}\alpha^{-1}{\,\mathrm{d}}\alpha \ll \log (2N), \end{align*}$$

and when $p>1$ , meanwhile,

$$\begin{align*}\int_0^1\bigg| \sum_{1\leqslant n\leqslant N}e(n\alpha)\bigg|^p{\,\mathrm{d}}\alpha \ll N^{p-1}+\int_{1/N}^{1/2}\alpha^{-p}{\,\mathrm{d}}\alpha \ll N^{p-1}. \end{align*}$$

The lower bound implicit in the first estimate of the theorem follows via a straightforward exercise using the formula (2.1), and so the conclusion of the theorem is complete.

Proof. There is no loss of generality in restricting attention to the scenario in which $\alpha>1$ . For if $0<\alpha \leqslant 1$ , then the set $\mathscr B(\alpha ,\beta )$ contains all large natural numbers, and the desired conclusion is essentially immediate from Theorem 2.1. Next, with our subsequent deliberations in mind, we equip ourselves with a conclusion concerning Diophantine approximations to the real number $1/\alpha $ . Since $1/\alpha $ has finite Diophantine type, we may suppose that there is a natural number k with the property that

$$\begin{align*}\liminf_{q\rightarrow \infty}q^k\| q/\alpha \|>0. \end{align*}$$

In particular, there exists a positive number $c=c(\alpha )$ having the property that, for all large enough natural numbers q, and for all integers a, one has

(2.2) $$ \begin{align} \Big| \frac{1}{\alpha}-\frac{a}{q}\Big|\geqslant \frac{c}{q^{k+1}}. \end{align} $$

Our plan is to adopt an argument relating exponential sums over Beatty sequences to corresponding unrestricted exponential sums, following a path close to that pursued in the proof of [Reference Banks, Güloğlu and Vaughan4, Lemma 4.3]. Along the way, we incorporate adjustments and minor corrections relative to the latter source. We begin by observing that $n=\lfloor \alpha m+\beta \rfloor $ for some $m\in \mathbb Z$ if and only if

(2.3) $$ \begin{align} 1-\alpha^{-1}<\{ \alpha^{-1}(n-\beta)\}<1. \end{align} $$

For if $\alpha ^{-1}(n-\beta )=m-1+\delta $ , for some $\delta \in [0,1]$ and $m\in \mathbb Z$ , then

$$\begin{align*}n=\alpha m+\beta -\alpha (1-\delta), \end{align*}$$

whence

$$\begin{align*}\lfloor \alpha m+\beta\rfloor =\lfloor n+\alpha (1-\delta)\rfloor. \end{align*}$$

This last integer is equal to n if and only if $0\leqslant \alpha (1-\delta )<1$ . Thus, we have $\lfloor \alpha m+\beta \rfloor =n$ if and only if $1-1/\alpha <\delta \leqslant 1$ , or equivalently, one has

$$\begin{align*}1-1/\alpha <\{ m-1+\delta \}\leqslant 1. \end{align*}$$

The situation with $\delta =1-1/\alpha $ corresponds to $n=\alpha m+\beta -1$ , whilst that with $\delta =1$ corresponds to $n=\alpha m+\beta $ . Since $\alpha $ is irrational, these scenarios can happen in their respective situations at most once.

Next, define the function $\psi :\mathbb R\rightarrow [-1/2,1/2)$ by putting

$$\begin{align*}\psi(x)=x-\lfloor x\rfloor -1/2, \end{align*}$$

and observe that $\psi (x)$ is periodic with period $1$ . When $x\in [0,1)$ , moreover, one has

$$\begin{align*}\alpha^{-1}+\psi(x)-\psi(x+1/\alpha)=\begin{cases}1,&\text{when } 1-1/\alpha\leqslant x<1,\\ 0,&\text{when } 0\leqslant x<1-1/\alpha.\end{cases} \end{align*}$$

Here, we have corrected the corresponding statement in the proof of [Reference Banks, Güloğlu and Vaughan4, Lemma 4.3]. It therefore follows from the criterion (2.3) that

(2.4) $$ \begin{align} \sum_{\substack{1\leqslant n\leqslant N\\ n\in \mathscr B(\alpha, \beta)}}e(n\theta)=S_1(\theta)+S_2(\theta), \end{align} $$

where

$$\begin{align*}S_1(\theta)=\frac{1}{\alpha}\sum_{1\leqslant n\leqslant N}e(n\theta) \end{align*}$$

and

$$\begin{align*}S_2(\theta)=\sum_{1\leqslant n\leqslant N}\Big( \psi \Big( \frac{n-\beta}{\alpha}\Big) -\psi\Big( \frac{n+1-\beta}{\alpha}\Big) \Big) e(\theta n)+O(1). \end{align*}$$

The most difficult part of the decomposition (2.4) concerns the expression $S_2(\theta )$ . Again following [Reference Banks, Güloğlu and Vaughan4], we write

$$\begin{align*}W(\varphi)=\sum_{1\leqslant n\leqslant N}\min\{ 1,N^{-k}\|n/\alpha -\varphi\|^{-1}\}. \end{align*}$$

Then, by applying Montgomery and Vaughan [Reference Montgomery and Vaughan22, Lemma D.1], we see that

(2.5) $$ \begin{align} S_2(\theta)=\,\sum_{0<|h|\leqslant N^k}&\frac{e(-\beta h/\alpha)-e((1-\beta )h/\alpha)}{2\pi\mathrm{i}h}\sum_{1\leqslant n\leqslant N} e(n(\theta +h/\alpha))\notag \\ &\, +O(1+W((\beta-1)/\alpha)+W(\beta/\alpha)). \end{align} $$

By Dirichlet’s approximation theorem, there exist $a\in \mathbb Z$ and $q\in \mathbb N$ with $(a,q)=1$ and $1\leqslant q\leqslant N^k$ satisfying the bound $|q/\alpha -a|\leqslant N^{-k}$ . Since $\alpha $ has the property (2.2), we find that

$$\begin{align*}cq^{-k-1}\leqslant \Big| \frac{1}{\alpha}-\frac{a}{q}\Big|\leqslant q^{-1}N^{-k}, \end{align*}$$

whence $q^k\geqslant cN^k$ , and in particular $q\gg N$ . We therefore deduce from Baker [Reference Baker2, Lemma 3.2] that

$$ \begin{align*} W(\varphi)&=N^{-k}\sum_{1\leqslant n\leqslant N}\min\{ N^k,\|n/\alpha -\varphi\|^{-1}\}\\ &\ll N^{-k}(N^k+q\log q)(N/q+1)\\ &\ll N\left( q^{-1}+N^{-1}+(q\log q)N^{-k-1}+(\log q)N^{-k}\right). \end{align*} $$

Since we may suppose that $N\ll q\leqslant N^k$ , we therefore see that $W(\varphi )\ll \log (2N)$ , and hence we deduce from (2.5) that

(2.6) $$ \begin{align} S_2(\theta)\ll \log (2N)+\sum_{0<|h|\leqslant N^k}\frac{1}{|h|}\min \{ N, \|\theta +h/\alpha\|^{-1}\}. \end{align} $$

By a change of variable, we see that

$$ \begin{align*} \int_0^1|S_2(\theta)|{\,\mathrm{d}}\theta &\ll \log (2N)+\sum_{0<|h|\leqslant N^k}\frac{1}{|h|}\int_0^1\min\{N,\|\theta\|^{-1}\} {\,\mathrm{d}}\theta \\ &\ll \log (2N)+\sum_{0<|h|\leqslant N^k}\frac{1}{|h|}\log (2N), \end{align*} $$

whence

$$\begin{align*}\int_0^1|S_2(\theta)|{\,\mathrm{d}}\theta \ll \log^2(2N). \end{align*}$$

We therefore deduce from (2.4) and the argument associated with Theorem 2.1 that

$$ \begin{align*} \int_0^1\bigg| \sum_{\substack{1\leqslant n\leqslant N\\ n\in \mathscr B(\alpha,\beta)}}e(n\theta)\bigg|{\,\mathrm{d}}\theta &\leqslant \frac{1}{\alpha}\int_0^1\bigg| \sum_{1\leqslant n\leqslant N}e(n\theta)\bigg|{\,\mathrm{d}}\theta +\int_0^1|S_2(\theta)|{\,\mathrm{d}}\theta \\ &\ll \frac{1}{\alpha}\log (2N)+\log^2(2N). \end{align*} $$

The first conclusion of the theorem is now immediate.

The second bound of the theorem requires additional effort. We suppose throughout that p is a real number with $p>1$ . We observe first that from (2.6), there is a choice of $\eta \in \{ +1,-1\}$ for which

$$\begin{align*}S_2(\theta)\ll\log (2N)+\sum_{1\leqslant h\leqslant N^k}\frac{1}{h}\min\{ N,\|h/\alpha +\eta \theta\|^{-1}\}. \end{align*}$$

By dividing the summation over h into dyadic intervals, we find that

(2.7) $$ \begin{align} S_2(\theta)\ll \log (2N)+\sum_{j=0}^J 2^{-j}U(2^j), \end{align} $$

where $J=\lfloor k(\log N)/(\log 2)\rfloor $ , and

(2.8) $$ \begin{align} U(H)=\sum_{H\leqslant h\leqslant 2H}\min\{ N,\|h/\alpha +\eta \theta\|^{-1}\}. \end{align} $$

Consider a typical value of $H=2^j$ with $0\leqslant j\leqslant J$ . By Dirichlet’s approximation theorem, there exist $a\in \mathbb Z$ and $q\in \mathbb N$ with $(a,q)=1$ and $1\leqslant q\leqslant H$ satisfying the bound

$$\begin{align*}|q/\alpha -a|\leqslant H^{-1}. \end{align*}$$

Since $\alpha $ has the property (2.2), we find that

$$\begin{align*}cq^{-k-1}\leqslant \Big| \frac{1}{\alpha}-\frac{a}{q}\Big|\leqslant q^{-1}H^{-1}, \end{align*}$$

whence $q\gg H^{1/k}$ . We therefore deduce from Baker [Reference Baker2, Lemma 3.2] that

$$ \begin{align*} \sum_{H\leqslant h\leqslant 2H}\min\{ N,\|h/\alpha +\eta \theta\|^{-1}\}&\ll (N+q\log q)(H/q+1)\\ &\ll NH\Big( \frac{1}{q}+\frac{1}{H}+\frac{q\log q}{NH}+\frac{\log q}{N}\Big). \end{align*} $$

Since we may suppose that $H^{1/k}\ll q\leqslant H\ll N^k$ , we discern that

(2.9) $$ \begin{align} U(H)=\sum_{H\leqslant h\leqslant 2H}\min\{ N,\| h/\alpha +\eta \theta\|^{-1}\} \ll NH^{1-1/k}\log (2H)\ll NH^{1-1/(2k)}. \end{align} $$

By applying Hölder’s inequality and substituting the estimate (2.9), we find from (2.8) that

$$ \begin{align*} U(H)^p&\ll U(H)^{(p-1)/2}H^{(p-1)/2}\sum_{H\leqslant h\leqslant 2H}\left( \min\{ N,\|h/\alpha +\eta \theta\|^{-1}\} \right)^{(p+1)/2}\\ &\ll (NH^{1-1/(2k)})^{(p-1)/2}H^{(p-1)/2}\sum_{H\leqslant h\leqslant 2H}\left( \min\{ N,\|h/\alpha +\eta \theta\|^{-1}\} \right)^{(p+1)/2}. \end{align*} $$

Since we are assuming throughout that $p>1$ , it follows via a change of variable that

(2.10) $$ \begin{align} \int_0^1 U(H)^p{\,\mathrm{d}}\theta &\ll N^{(p-1)/2}H^{p-1-(p-1)/(4k)}\sum_{H\leqslant h\leqslant 2H}\int_0^1 \left( \min\{ N,\|\beta\|^{-1}\} \right)^{(p+1)/2}{\,\mathrm{d}}\beta \notag \\ &\ll N^{(p-1)/2}H^{p-(p-1)/(4k)}N^{(p+1)/2-1}\notag \\ &\ll N^{p-1}H^{p-2\delta (p-1)}, \end{align} $$

where we write $\delta =1/(8k)$ .

We now return to (2.7), observing that an additional application of Hölder’s inequality reveals that

$$\begin{align*}|S_2(\theta)|^p\ll \log^p(2N)+\bigg( \sum_{j=0}^J2^{-\delta j}\bigg)^{p-1}\sum_{j=0}^J 2^{(\delta (p-1)-p)j} U(2^j)^p. \end{align*}$$

By applying (2.10), we therefore infer that

$$ \begin{align*} \int_0^1 |S_2(\theta )|^p{\,\mathrm{d}}\theta &\ll \log^p(2N)+\sum_{j=0}^J 2^{(\delta (p-1)-p)j}\int_0^1 U(2^j)^p{\,\mathrm{d}}\theta \\ &\ll \log^p(2N)+\sum_{j=0}^J 2^{(\delta(p-1)-p)j}N^{p-1}(2^j)^{p-2\delta (p-1)}\\ &\ll \log^p(2N)+N^{p-1}\sum_{j=0}^J2^{-\delta (p-1)j}\ll N^{p-1}. \end{align*} $$

Finally, on substituting this bound into a consequence of the decomposition (2.4), we arrive at the upper bound

$$ \begin{align*} \int_0^1 \bigg| \sum_{\substack{1\leqslant n\leqslant N\\ n\in \mathscr B(\alpha,\beta)}}e(n\theta )\bigg|^p{\,\mathrm{d}}\theta &\ll \int_0^1 |S_1(\theta)|^p{\,\mathrm{d}}\theta +\int_0^1|S_2(\theta)|^p{\,\mathrm{d}}\theta \\ &\ll \int_0^1 \bigg| \sum_{1\leqslant n\leqslant N}e(n\theta )\bigg|^p{\,\mathrm{d}}\theta +N^{p-1}. \end{align*} $$

The second conclusion of the theorem is now immediate from the second estimate of Theorem 2.1. This completes the proof of the theorem.

Proof. One has

$$\begin{align*}\sum_{n\in \mathscr C(N)}e(n\alpha )=\sum_{n\in \mathscr B(N)}e(n\alpha) -\sum_{n\in \mathscr A(N)}e(n\alpha). \end{align*}$$

It therefore follows from (1.1) and (1.3) that

$$\begin{align*}I_p(N;\mathscr C)\ll I_p(N;\mathscr B)+I_p(N;\mathscr A)\ll N^{p-1}. \end{align*}$$

Hence, provided that $\mathscr C$ has positive lower density, we see that it is a strongly subconvex $L^p$ -set.

Proof. The set $\mathbb N$ is a strongly subconvex $L^q$ -set whenever $q>1$ , and hence also a strongly subconvex $L^p$ -set. Thus, the conclusion is immediate from Theorem 3.1.

Proof. As noted in Example 2.2, both $\mathscr N_r$ and $\mathscr N_s$ are strongly subconvex $L^p$ -sets whenever $p>1+1/s$ . Since $\mathscr N_s\subset \mathscr N_r$ and $\mathscr N_r\setminus \mathscr N_s$ has positive density, the desired conclusion follows from Theorem 3.1.

Proof. We have

$$\begin{align*}\sum_{n\in \mathscr B(N)}e(n\alpha)=\sum_{n\in \mathscr A(N/q)}e((qn+a)\alpha)+O(1), \end{align*}$$

whence, by a change of variable, it follows from (1.1) that

$$ \begin{align*} I_p(N;\mathscr B)&\ll 1+\int_0^1\bigg| \sum_{n\in \mathscr A(N/q)}e(qn\alpha)\bigg|^p{\,\mathrm{d}}\alpha \\ &\ll 1+I_p(N/q;\mathscr A)\ll N^{p-1}. \end{align*} $$

The desired conclusion follows at once.

Proof. One has

$$\begin{align*}\sum_{n\in \mathscr A'(N)}e(n\alpha)=\sum_{n\in \mathscr A(N)}e(n\alpha)-\sum_{n\in \mathscr B(N)}e(n\alpha)+ \sum_{n\in \mathscr C(N)}e(n\alpha), \end{align*}$$

whence, it follows from (1.1) that

$$\begin{align*}I_p(N;\mathscr A')\ll I_p(N;\mathscr A)+I_p(N;\mathscr B)+I_p(N;\mathscr C). \end{align*}$$

Since we may assume that $p<2$ , it follows from Hölder’s inequality and orthogonality that one has

$$\begin{align*}I_p(N;\mathscr B)\leqslant I_2(N;\mathscr B)^{p/2}\leqslant B(N)^{p/2}, \end{align*}$$

with a similar upper bound available for the mean value over the exponential sum associated with the set $\mathscr C(N)$ . Since $\mathscr A$ is a strongly subconvex $L^p$ -set, it therefore follows from the hypothesis (3.1) that

$$ \begin{align*} I_p(N;\mathscr A')&\ll N^{p-1}+B(N)^{p/2}+C(N)^{p/2}\\ &\ll N^{p-1}+(N^{2-2/p})^{p/2}\ll N^{p-1}. \end{align*} $$

This confirms that $\mathscr A'$ is a strongly subconvex $L^p$ -set.

Proof. The desired conclusion concerning $\mathscr A\cap \mathscr B$ is essentially a consequence of Young’s convolution inequality (see [Reference Brascamp and Lieb7], for example), though the argument is sufficiently simple and instructive that we take the opportunity to describe our proof in detail. We recall that as a consequence of Definition 1.1, it is implicit that the strong subconvexity of $\mathscr A$ and $\mathscr B$ implies that $q<2$ and $r<2$ . Define the real number p via the relation (3.3), and note that the condition (3.4) then ensures that $1<p<2$ . Also, put $\mathscr C=\mathscr A\cap \mathscr B$ . Then on making use of the notation (3.2), it follows from orthogonality that one has

$$\begin{align*}f(\alpha;\mathscr C)=\int_0^1 f(\alpha-\beta;\mathscr A)f(\beta ;\mathscr B){\,\mathrm{d}}\beta. \end{align*}$$

Next, observe that as a consequence of the last relation, it follows from (1.1) that

$$ \begin{align*} I_p(N;\mathscr C)&=\int_0^1 |f(\alpha;\mathscr C)|^{p-1}\Big| \int_0^1 f(\alpha-\beta;\mathscr A) f(\beta;\mathscr B){\,\mathrm{d}}\beta \Big| {\,\mathrm{d}}\alpha \\ &\leqslant \int_0^1 \int_0^1 |f(\alpha;\mathscr C)|^{p-1}|f(\alpha-\beta;\mathscr A)f(\beta;\mathscr B)|{\,\mathrm{d}}\beta {\,\mathrm{d}}\alpha. \end{align*} $$

An application of Hölder’s inequality therefore reveals that

(3.5) $$ \begin{align} I_p(N;\mathscr C)\leqslant T_1^{\frac{1}{q}-\frac{1}{p}}T_2^{\frac{1}{r}-\frac{1}{p}}T_3^{\frac{1}{p}}, \end{align} $$

where

$$ \begin{align*} T_1&=\int_0^1\int_0^1 |f(\alpha;\mathscr C)|^p|f(\alpha-\beta;\mathscr A)|^q{\,\mathrm{d}}\beta {\,\mathrm{d}}\alpha ,\\ T_2&=\int_0^1\int_0^1 |f(\alpha;\mathscr C)|^p|f(\beta;\mathscr B)|^r{\,\mathrm{d}}\beta {\,\mathrm{d}}\alpha ,\\ T_3&=\int_0^1\int_0^1 |f(\alpha-\beta;\mathscr A)|^q|f(\beta;\mathscr B)|^r{\,\mathrm{d}}\beta {\,\mathrm{d}}\alpha. \end{align*} $$

Here, we have made use of the observation that

$$\begin{align*}\frac{1}{q}-\frac{1}{p}=1-\frac{1}{r}, \end{align*}$$

so that $0<1/q-1/p<1$ , and similarly $0<1/r-1/p<1$ . Moreover,

$$\begin{align*}\Big( \frac{1}{q}-\frac{1}{p}\Big) +\Big( \frac{1}{r}-\frac{1}{p}\Big) +\frac{1}{p}=\frac{1}{q}+\frac{1}{r}-\frac{1}{p}=1. \end{align*}$$

Hence, our application of Hölder’s inequality is legitimate.

By applying a change of variables, and utilising the definition (1.1) and the fact that $\mathscr A$ is a strongly subconvex $L^q$ -set, we find that

$$\begin{align*}T_1=I_p(N;\mathscr C)I_q(N;\mathscr A)\ll N^{q-1}I_p(N;\mathscr C). \end{align*}$$

Similarly, one sees that $T_2\ll N^{r-1}I_p(N;\mathscr C)$ , and

$$\begin{align*}T_3=I_q(N;\mathscr A)I_r(N;\mathscr B)\ll N^{q+r-2}. \end{align*}$$

We therefore deduce from (3.5) that

(3.6) $$ \begin{align} I_p(N;\mathscr C)&\ll \left( N^{q-1}I_p(N;\mathscr C)\right)^{\frac{1}{q}-\frac{1}{p}} \left( N^{r-1}I_p(N;\mathscr C)\right)^{\frac{1}{r}-\frac{1}{p}} \left( N^{q+r-2}\right)^{1/p}\notag \\ &\ll N^{1-1/p}I_p(N;\mathscr C)^{1-1/p}. \end{align} $$

Our hypothesis that the set $\mathscr C=\mathscr A\cap \mathscr B$ has positive lower density ensures that when $0\leqslant \alpha \leqslant (100N)^{-1}$ , one has

$$\begin{align*}|f(\alpha;\mathscr C)|=\bigg| \sum_{n\in \mathscr C(N)}e(n\alpha)\bigg|\geqslant \frac{1}{2}|C(N)|\gg N, \end{align*}$$

whence

$$\begin{align*}I_p(N;\mathscr C)\geqslant \int_0^{1/(100N)}|f(\alpha ;\mathscr C)|^p{\,\mathrm{d}}\alpha \gg N^{p-1}. \end{align*}$$

Then we conclude from (3.6) that $I_p(N;\mathscr C)^{1/p}\ll N^{1-1/p}$ , whence $I_p(N;\mathscr C)\ll N^{p-1}$ . This confirms that $\mathscr A\cap \mathscr B$ is a strongly subconvex $L^p$ -set.

The conclusion concerning $\mathscr A\cup \mathscr B$ follows by observing that

$$\begin{align*}f(\alpha;\mathscr A\cup \mathscr B)=f(\alpha;\mathscr A)+f(\alpha;\mathscr B)-f(\alpha;\mathscr A\cap \mathscr B). \end{align*}$$

Since our work on the case $\mathscr A\cap \mathscr B$ shows that the p-th moment of each of the exponential sums on the right hand side is $O(N^{p-1})$ , the desired conclusion $I_p(N;\mathscr A\cup \mathscr B)\ll N^{p-1}$ is immediate.

Proof. The Beatty sequence $\mathscr B(\alpha ,\beta )$ is a strongly subconvex $L^r$ -set whenever $1<r<2$ , as a consequence of Theorem 2.3. Choose any such value of r sufficiently close to $1$ , and define the real number p by means of the relation (3.3). Then we may take p as close as we like to q by choosing r suitably close to $1$ . The conclusion of Theorem 3.6 then shows that $\mathscr A\cap \mathscr B(\alpha ,\beta )$ is a strongly subconvex $L^p$ -set provided that this set has positive lower density and condition (3.4) is satisfied. Since $q>1$ and $r>1$ , the upper bound on $1/q+1/r$ in the latter condition is automatically satisfied. Moreover, one has $q<2$ , so the lower bound $1/q+1/r>3/2$ is satisfied on taking r sufficiently close to $1$ . The desired conclusion therefore follows.

Proof of Theorem 1.2.

Let $\mathscr A$ be a weakly subconvex $L^p$ -set for some real number p with $1\leqslant p\leqslant 4/3$ . We may suppose that $k\geqslant 3$ , and that $a\in \mathbb Z$ and $q\in \mathbb N$ satisfy $(a,q)=1$ and $|\alpha _k-a/q|\leqslant q^{-2}$ . We recall the definition (1.6), and introduce the exponential sum $F_k(\boldsymbol \alpha )=F_k(\boldsymbol \alpha ;N)$ , defined by

(4.1) $$ \begin{align} F_k(\boldsymbol \alpha;N)=\sum_{n\in \mathscr A(N)}e(\psi(n;\boldsymbol \alpha)). \end{align} $$

We also make use of the exponential sum $g(\alpha )=f_N(\alpha ;\mathscr A)$ defined via (3.2), so that

(4.2) $$ \begin{align} g(\alpha)=\sum_{n\in \mathcal A(N)}e(n\alpha), \end{align} $$

and introduce the exponential sum

(4.3) $$ \begin{align} G_k(\boldsymbol \alpha ,\beta)=\sum_{1\leqslant n\leqslant N}e(\psi(n;\boldsymbol \alpha )+\beta n). \end{align} $$

We note that $G_k(\boldsymbol \alpha ,\beta )$ is merely a more convenient form of an exponential sum that may be written in terms of the sum $\Psi _k(\boldsymbol \alpha ;N)$ defined in (1.7). It now follows from orthogonality that one has the fundamental relation

(4.4) $$ \begin{align} F_k(\boldsymbol \alpha)=\int_0^1G_k(\boldsymbol \alpha,\beta)g(-\beta){\,\mathrm{d}}\beta. \end{align} $$

An application of Hölder’s inequality reveals that

(4.5) $$ \begin{align} |F_k(\boldsymbol \alpha)|\leqslant \Big( \int_0^1|G_k(\boldsymbol \alpha,\beta)|^4{\,\mathrm{d}}\beta \Big)^{1/4} \Big( \int_0^1|g(\beta )|^{4/3}{\,\mathrm{d}}\beta \Big)^{3/4}. \end{align} $$

Here, our hypothesis that $\mathscr A$ is a weakly subconvex $L^p$ -set for some real number p with $p\leqslant 4/3$ ensures that

$$\begin{align*}\int_0^1|g(\beta)|^{4/3}{\,\mathrm{d}}\beta \leqslant N^{4/3-p}\int_0^1|g(\beta )|^p{\,\mathrm{d}}\beta \ll N^{1/3+\varepsilon}. \end{align*}$$

Hence, we deduce from (4.5) that

(4.6) $$ \begin{align} |F_k(\boldsymbol \alpha)|^4\ll N^{1+\varepsilon}\int_0^1|G_k(\boldsymbol \alpha,\beta)|^4{\,\mathrm{d}}\beta. \end{align} $$

By orthogonality, the integral on the right hand side of (4.6) may be interpreted in the form

$$\begin{align*}\int_0^1|G_k(\boldsymbol \alpha ,\beta)|^4{\,\mathrm{d}}\beta =\sum_{\substack{1\leqslant n_1,n_2,n_3,n_4\leqslant N\\ n_1+n_2=n_3+n_4 }}e(\psi(n_1;\boldsymbol \alpha)+\psi(n_2;\boldsymbol \alpha)- \psi(n_3;\boldsymbol \alpha)-\psi(n_4;\boldsymbol \alpha)). \end{align*}$$

For suitable integers $h_1$ and $h_2$ with $|h_i|<N\, (i=1,2)$ , one can write the summands $n_3$ and $n_4$ in the shape

$$\begin{align*}n_3=n_1+h_1\quad \text{and}\quad n_4=n_1+h_2. \end{align*}$$

The relation $n_1+n_2=n_3+n_4$ then implies that $n_2=n_1+h_1+h_2$ . Thus, making use of the forward difference operator

$$\begin{align*}{\Delta}_1(\varphi(x);h)=\varphi(x+h)-\varphi(x), \end{align*}$$

and the second order operator

$$\begin{align*}{\Delta}_2(\varphi(x);h_1,h_2)={\Delta}_1({\Delta}_1(\varphi(x);h_1);h_2), \end{align*}$$

we see that

$$\begin{align*}\psi(n_1;\boldsymbol \alpha)+\psi(n_2;\boldsymbol \alpha)-\psi(n_3;\boldsymbol \alpha)- \psi(n_4;\boldsymbol \alpha)={\Delta}_2(\psi(n_1;\boldsymbol \alpha);h_1,h_2). \end{align*}$$

In this way, we conclude that

(4.7) $$ \begin{align} \int_0^1|G_k(\boldsymbol \alpha ,\beta)|^4{\,\mathrm{d}}\beta =\sum_{|h_1|<N}\sum_{|h_2|<N}\sum_{n\in \mathscr I(N;{\mathbf h})} e(\Delta_2(\psi(n;\boldsymbol \alpha );h_1,h_2)), \end{align} $$

in which $\mathscr I(N;{\mathbf h})$ denotes the interval of integers defined by

$$\begin{align*}\mathscr I(N;{\mathbf h})=[1,N]\cap [1-h_1,N-h_1]\cap [1-h_2,N-h_2]\cap [1-h_1-h_2,N-h_1-h_2]. \end{align*}$$

The right hand side of (4.7) will be recognised as the principal output of a second order Weyl differencing process applied to the exponential sum $G_k(\boldsymbol \alpha ,0)=\Psi _k(\boldsymbol \alpha ;N)$ . The interested reader should direct their attention to [Reference Vaughan24, Lemma 2.3] and compare with the situation therein with $j=2$ .

On substituting (4.7) into (4.6), we see that

$$\begin{align*}|F_k(\boldsymbol \alpha)|^{2^2}\ll N^{1+\varepsilon}\sum_{|h_1|<N}\sum_{|h_2|<N}T_2, \end{align*}$$

where

$$\begin{align*}T_2=\sum_{x\in \mathscr I_2}e(\Delta_2(\psi(x;\boldsymbol \alpha);h_1,h_2)), \end{align*}$$

in which $\mathscr I_2=\mathscr I(N;{\mathbf h})$ . A comparison of this upper bound with the statement of [Reference Vaughan24, Lemma 2.3] reveals, just as in the proof of [Reference Vaughan24, Lemma 2.4], that the application of $k-3$ further Weyl differencing steps delivers the bound

$$\begin{align*}|F_k(\boldsymbol \alpha)|^{2^{k-1}}\ll N^{2^{k-1}-k+\varepsilon}\sum_{|h_1|<N}\cdots \sum_{|h_{k-1}|<N} \sum_{x\in \mathscr I_{k-1}}e(h_1\cdots h_{k-1}p_{k-1}(x;h_1,\ldots ,h_{k-1})), \end{align*}$$

where $\mathscr I_{k-1}$ is a subinterval of integers lying within $[1,N]$ , and

$$\begin{align*}p_{k-1}(x;h_1,\ldots ,h_{k-1})=k!\alpha_k(x+\tfrac{1}{2}h_1+\ldots+\tfrac{1}{2}h_{k-1})+(k-1)!\alpha_{k-1}. \end{align*}$$

Hence, just as in the completion of the argument of the proof of [Reference Vaughan24, Lemma 2.4], one concludes that

$$\begin{align*}|F_k(\boldsymbol \alpha)|^{2^{k-1}}\ll N^{2^{k-1}+\varepsilon}(q^{-1}+N^{-1}+qN^{-k}). \end{align*}$$

Thus

$$\begin{align*}|F_k(\boldsymbol \alpha )|\ll N^{1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{2^{1-k}}, \end{align*}$$

confirming the desired conclusion (1.9).

Proof. We begin by noting that when $\mathscr A$ is a weakly subconvex $L^1$ -set, then it is also a weakly subconvex $L^p$ -set for any $p>1$ . The conclusion of Theorem 1.3 for $p=1$ is obtained from that for $1<p<2$ by taking p sufficiently close to $1$ , on adjusting the value of $\varepsilon $ in the desired conclusion. There is consequently no loss of generality in supposing throughout that $1<p<2$ and $k\geqslant 2$ . Next, we observe that in the case $k=2$ it makes little sense to perform two Weyl differencing steps, so that the proof of Theorem 1.2 is of limited use to us. In this case we nonetheless have the orthogonality relation (4.4), but in present circumstances we apply Hölder’s inequality to obtain the upper bound

(4.9) $$ \begin{align} |F_k(\boldsymbol \alpha)|\leqslant \Big( \int_0^1|G_k(\boldsymbol \alpha, \beta)|^{p/(p-1)}{\,\mathrm{d}}\beta \Big)^{1-1/p} \Big( \int_0^1|g(\beta)|^p{\,\mathrm{d}}\beta \Big)^{1/p}. \end{align} $$

Our hypothesis that $\mathcal A$ is a weakly subconvex $L^p$ -set ensures via (1.2) that

(4.10) $$ \begin{align} \int_0^1|g(\beta)|^p{\,\mathrm{d}}\beta \ll N^{p-1+\varepsilon}. \end{align} $$

Moreover, since $1<p<2$ , we may suppose that $p/(p-1)>2$ . Hence, by orthogonality, we see that

(4.11) $$ \begin{align} \int_0^1|G_k(\boldsymbol \alpha,\beta)|^{\frac{p}{p-1}}{\,\mathrm{d}}\beta &\leqslant \Big( \sup_{\beta\in [0,1)}|G_k(\boldsymbol \alpha,\beta)|\Big)^{\frac{2-p}{p-1}} \int_0^1 |G_k(\boldsymbol \alpha,\beta)|^2{\,\mathrm{d}}\beta \notag \\ & \leqslant N\Big( \sup_{\beta\in [0,1)}|G_k(\boldsymbol \alpha ,\beta)|\Big)^{\frac{2-p}{p-1}}. \end{align} $$

Since $a\in \mathbb Z$ and $q\in \mathbb N$ satisfy $(a,q)=1$ and $|\alpha _k-a/q|\leqslant q^{-2}$ , it follows from Weyl’s inequality (see [Reference Vaughan24, Lemma 2.4]) that we have the bound

$$\begin{align*}G_k(\boldsymbol \alpha ,\beta)\ll N^{1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{2^{1-k}}. \end{align*}$$

Thus we see from (4.11) that

$$\begin{align*}\int_0^1|G_k(\boldsymbol \alpha,\beta)|^{\frac{p}{p-1}}{\,\mathrm{d}}\beta \ll N\cdot N^{\frac{2-p}{p-1}+\varepsilon} (q^{-1}+N^{-1}+qN^{-k})^{\left( \frac{2-p}{p-1}\right)2^{1-k}}. \end{align*}$$

On substituting this estimate together with (4.10) into (4.9), we conclude that

$$ \begin{align*} F_k(\boldsymbol \alpha)&\ll N^{\frac{1}{p}+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{\left( \frac{2}{p}-1\right)2^{1-k}} (N^{p-1+\varepsilon})^{\frac{1}{p}}\\ &\ll N^{1+2\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{\left( \frac{2}{p}-1\right) 2^{1-k}}. \end{align*} $$

Thus far, our discussion has been independent of the value of $k\geqslant 2$ . Specialising to the situation with $k=2$ , however, we obtain the estimate (4.8) with $\sigma _p(k)=\frac {1}{p}-\frac {1}{2}$ , confirming the conclusion of the lemma in this case.

The situation with $k\geqslant 3$ and $1\leqslant p\leqslant 4/3$ is immediate from Theorem 1.2, since, in such circumstances, one has $\sigma _p(k)=2^{1-k}$ . We turn now to address the situation in which $k\geqslant 3$ and $4/3<p<2$ . Here, our starting point is the relation (4.4), though we apply Hölder’s inequality in a manner that incorporates the $4$ -th moment of $G_k(\boldsymbol \alpha ,\beta )$ . Thus, we obtain

(4.12) $$ \begin{align} |F_k(\boldsymbol \alpha)|\leqslant U_1^{\frac{3}{2}-\frac{2}{p}}U_2^{\frac{1}{p}-\frac{1}{2}}\Big( \int_0^1 |g(\beta)|^p{\,\mathrm{d}}\beta \Big)^{1/p}, \end{align} $$

where

(4.13) $$ \begin{align} U_r=\int_0^1|G_k(\boldsymbol \alpha ,\beta)|^{2r}{\,\mathrm{d}}\beta \quad (r\geqslant 1). \end{align} $$

By orthogonality, one finds from (4.3) that $U_1\leqslant N$ . Meanwhile, the argument leading from (4.7) to the conclusion of the proof of Theorem 1.2 shows that

$$\begin{align*}U_2\ll N^{3+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{2^{3-k}}. \end{align*}$$

Thus, on recalling the upper bound (4.10), we deduce from (4.12) that

$$ \begin{align*} F_k(\boldsymbol \alpha)&\ll N^{\frac{3}{2}-\frac{2}{p}+\varepsilon}\left( N^3(q^{-1}+N^{-1}+qN^{-k})^{2^{3-k}} \right)^{\frac{1}{p}-\frac{1}{2}}(N^{p-1})^{\frac{1}{p}}\\ &\ll N^{1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{\left( \frac{1}{p}-\frac{1}{2}\right)2^{3-k}}. \end{align*} $$

Since $\sigma _p(k)=\big ( \frac {1}{p}-\frac {1}{2}\big )2^{3-k}$ when $k\geqslant 3$ and $4/3<p<2$ , this confirms the conclusion of the lemma in this final case.

Proof. Just as in the proof of Lemma 4.1, there is no loss of generality in supposing throughout that $1<p<2$ . Furthermore, in view of the definition of $\tau _p(k)$ , we may suppose in addition that $k\geqslant 3$ . Moreover, we again have the upper bounds (4.9) and (4.10) employed in the course of the proof of Lemma 4.1. We augment these estimates with a new mean value estimate. Write

(5.2) $$ \begin{align} r=\tfrac{1}{2}k(k-1). \end{align} $$

It transpires that we are able to bound $|F_k(\boldsymbol \alpha )|$ in terms of the mean value $U_r$ , defined as in (4.13). It is this mean value to which we apply Vinogradov’s methods.

Observe first that from (4.3), one has

$$\begin{align*}G_k(\boldsymbol \alpha ,\beta)= \frac{1}{N}\sum_{1\leqslant n\leqslant N}\sum_{1-n\leqslant h\leqslant N-n}e(\psi(n+h;\boldsymbol \alpha)+(n+h)\beta). \end{align*}$$

Thus, by Hölder’s inequality, one sees that

(5.3) $$ \begin{align} |G_k(\boldsymbol \alpha,\beta)|^{2r}\leqslant \frac{1}{N}\sum_{1\leqslant n\leqslant N}|E_k(\boldsymbol \alpha,\beta;n)|^{2r}, \end{align} $$

in which we write

$$\begin{align*}E_k(\boldsymbol \alpha,\beta;n)=\sum_{1-n\leqslant h\leqslant N-n}e(\psi(n+h;\boldsymbol \alpha)+h\beta). \end{align*}$$

Next, write

$$\begin{align*}\varphi(h;\boldsymbol \beta)=\beta_{k-2}h^{k-2}+\ldots +\beta_1h, \end{align*}$$

and put

$$\begin{align*}D_k(\boldsymbol \alpha ,\boldsymbol \beta;n)=\sum_{1-n\leqslant h\leqslant N-n} e(\psi(n+h;\boldsymbol \alpha)+\varphi(h;\boldsymbol \beta)). \end{align*}$$

Then we see via orthogonality that

$$\begin{align*}\int_0^1|E_k(\boldsymbol \alpha ,\beta;n)|^{2r}{\,\mathrm{d}}\beta =\sum_{|m_2|\leqslant 2rN^2}\cdots \sum_{|m_{k-2}|\leqslant 2rN^{k-2}} \int_{[0,1)^{k-2}}|D_k(\boldsymbol \alpha ,\boldsymbol \beta;n)|^{2r} e(-\boldsymbol \beta\cdot {\mathbf m}){\,\mathrm{d}}\boldsymbol \beta , \end{align*}$$

where we abbreviate $\beta _2m_2+\ldots +\beta _{k-2}m_{k-2}$ to $\boldsymbol \beta \cdot {\mathbf m}$ . An application of the triangle inequality therefore propels us from (5.3) to the bound

(5.4) $$ \begin{align} \int_0^1|G_k(\boldsymbol \alpha,\beta)|^{2r}{\,\mathrm{d}}\beta \ll \frac{1}{N}\sum_{1\leqslant n\leqslant N}N^{\frac{1}{2}(k-1)(k-2)-1} \int_{[0,1)^{k-2}}|D_k(\boldsymbol \alpha ,\boldsymbol \beta;n)|^{2r}{\,\mathrm{d}}\boldsymbol \beta. \end{align} $$

By orthogonality, the mean value on the right hand side of (5.4) is equal to

$$\begin{align*}\sum_{1-n\leqslant h_1\leqslant N-n}\cdots \sum_{1-n\leqslant h_{2r}\leqslant N-n}e(\sigma(n,{\mathbf h};\boldsymbol \alpha)), \end{align*}$$

in which we write

$$\begin{align*}\sigma(n,{\mathbf h};\boldsymbol \alpha)=\sum_{i=1}^r \left( \psi(n+h_i;\boldsymbol \alpha)-\psi(n+h_{r+i};\boldsymbol \alpha)\right) , \end{align*}$$

and the summation over $h_1,\ldots ,h_{2r}$ is subject to the condition

$$\begin{align*}\sum_{i=1}^r(h_i^j-h_{r+i}^j)=0\quad (1\leqslant j\leqslant k-2). \end{align*}$$

By expanding the polynomial $\psi (n+h;\boldsymbol \alpha )$ using the binomial theorem, moreover, we see that for each summand ${\mathbf h}$ , we have

$$\begin{align*}\sigma (n,{\mathbf h};\boldsymbol \alpha)=\alpha_k\sum_{i=1}^r(h_i^k-h_{r+i}^k)+(kn\alpha_k+\alpha_{k-1})\sum_{i=1}^r (h_i^{k-1}-h_{r+i}^{k-1}). \end{align*}$$

Hence, with the same implicit condition on $h_1,\ldots ,h_{2r}$ , we deduce that

(5.5) $$ \begin{align} \bigg| \sum_{1\leqslant n\leqslant N}\sum_{1-n\leqslant h_1\leqslant N-n}&\cdots \sum_{1-n\leqslant h_{2r}\leqslant N-n} e(\sigma(n,{\mathbf h};\boldsymbol \alpha))\bigg| \notag \\ &\leqslant \sum_{|h_1|\leqslant N}\cdots \sum_{|h_{2r}|\leqslant N}\bigg| \sum_{n\in \mathscr I({\mathbf h})} e\bigg( kn\alpha_k\sum_{i=1}^r(h_i^{k-1}-h_{r+i}^{k-1})\bigg) \bigg|, \end{align} $$

where $\mathscr I({\mathbf h})$ denotes the set of integers n in the interval $[1,N]$ satisfying the property that

$$\begin{align*}1-h_i\leqslant n\leqslant N-h_i\quad (1\leqslant i\leqslant 2r). \end{align*}$$

Since the set $\mathscr I({\mathbf h})$ is a subinterval of the integers lying in $[1,N]$ , it follows in a standard manner that by summing the implicit geometric progression, one has

$$\begin{align*}\sum_{n\in \mathscr I({\mathbf h})}e\bigg( kn\alpha_k\sum_{i=1}^r(h_i^{k-1}-h_{r+i}^{k-1})\bigg) \ll \min\bigg\{ N,\bigg\| k\alpha_k\sum_{i=1}^r (h_i^{k-1}-h_{r+i}^{k-1})\bigg\|^{-1}\bigg\}. \end{align*}$$

By substituting this bound into (5.5), and thence into (5.4), we conclude thus far that

(5.6) $$ \begin{align} \int_0^1|G_k(\boldsymbol \alpha ,\beta)|^{2r}{\,\mathrm{d}}\beta \ll N^{\frac{1}{2}(k-1)(k-2)-2}\sum_{|m_{k-1}|\leqslant 2rN^{k-1}} \rho(m_{k-1})\min\{ N,\|km\alpha_k\|^{-1}\}, \end{align} $$

where we write $\rho (m)$ for the number of solutions of the system of equations

$$ \begin{align*} \sum_{i=1}^r(h_i^{k-1}-h_{r+i}^{k-1})&=m,\\ \sum_{i=1}^r(h_i^j-h_{r+i}^j)&=0\quad (1\leqslant j\leqslant k-2), \end{align*} $$

with $|h_i|\leqslant N\, (1\leqslant i\leqslant 2r)$ .

A standard application of the recent resolution of the main conjecture on Vinogradov’s mean value theorem (see [Reference Bourgain, Demeter and Guth6], [Reference Wooley28], [Reference Wooley29]) shows via orthogonality that

$$ \begin{align*} \rho(m)&=\int_{[0,1)^{k-1}}\bigg| \sum_{|h|\leqslant N}e(\gamma_1h+\ldots +\gamma_{k-1}h^{k-1})\bigg|^{2r} e(-\gamma_{k-1}m){\,\mathrm{d}}\boldsymbol \gamma \\ &\ll 1+\int_{[0,1)^{k-1}}\bigg| \sum_{1\leqslant h\leqslant N}e(\gamma_1h+\ldots +\gamma_{k-1}h^{k-1})\bigg|^{2r} {\,\mathrm{d}}\boldsymbol \gamma \\ &\ll N^{r+\varepsilon}+N^{2r-\frac{1}{2}k(k-1)}. \end{align*} $$

Thus, we infer from (5.2) and (5.6) that

$$\begin{align*}\int_0^1 |G_k(\boldsymbol \alpha ,\beta)|^{2r}{\,\mathrm{d}}\beta \ll N^{2r-k-1+\varepsilon}\sum_{|m_{k-1}|\leqslant 2rN^{k-1}} \min\{ N,\|km\alpha_k\|^{-1}\}. \end{align*}$$

A standard reciprocal sums lemma (see, for example, [Reference Vaughan24, Lemma 2.2]) leads from here to the estimate

(5.7) $$ \begin{align} \int_0^1 |G_k(\boldsymbol \alpha,\beta)|^{2r}{\,\mathrm{d}}\beta \ll N^{2r-1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k}). \end{align} $$

We may now return to combine (5.7) with the bounds (4.9) and (4.10) already established. Observe first that when $p/(p-1)=2r$ , which is to say that $p=2r/(2r-1)$ , the estimate (5.7) may be substituted into (4.9) along with (4.10) to give

$$ \begin{align*} F_k(\boldsymbol \alpha)&\ll N^\varepsilon \left( N^{2r-1}(q^{-1}+N^{-1}+qN^{-k})\right)^{1/(2r)}(N^{p-1})^{1/p}\\ &\ll N^{1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{1/(2r)}. \end{align*} $$

Since any weakly subconvex $L^q$ -set with $q<2r/(2r-1)$ is also a weakly subconvex $L^p$ -set with $p=2r/(2r-1)$ , and

$$\begin{align*}\frac{2r}{2r-1}=1+\frac{1}{k^2-k-1}, \end{align*}$$

this establishes (5.1) when $k\geqslant 3$ and $1\leqslant p\leqslant 1+1/(k^2-k-1)$ .

The situation with $1+1/(k^2-k-1)<p<2$ requires more care. Applying Hölder’s inequality to (4.4), we obtain the upper bound

(5.8) $$ \begin{align} |F_k(\boldsymbol \alpha)|\leqslant U_1^{\omega_1}U_r^{\omega_r}\bigg( \int_0^1|g(\beta )|^p{\,\mathrm{d}}\beta \bigg)^{1/p}, \end{align} $$

where $U_1$ and $U_r$ are defined as in (4.13), and

$$\begin{align*}\omega_r=\Big( \frac{2}{p}-1\Big)\frac{1}{k^2-k-2}\quad \text{and}\quad \omega_1=1-\frac{1}{p}-\omega_r. \end{align*}$$

Notice here that since

$$\begin{align*}\frac{k^2-k}{k^2-k-1}<p<2, \end{align*}$$

one has

$$\begin{align*}0<\frac{2}{p}-1<\frac{k^2-k-2}{k^2-k}\quad \text{and}\quad \frac{1}{k^2-k}<1-\frac{1}{p}<1. \end{align*}$$

It follows that $\omega _r<1-1/p$ , whence our application of Hölder’s inequality was indeed legitimate.

The bound $U_1\leqslant N$ again follows via orthogonality, while the bound (5.7) supplies an estimate for $U_r$ . Consequently, on substituting these estimates along with (4.10) into (5.8), we deduce that

$$ \begin{align*} F_k(\boldsymbol \alpha)&\ll N^\varepsilon (N)^{\omega_1}(N^{2r-1}(q^{-1}+N^{-1}+qN^{-k}))^{\omega_r} (N^{p-1})^{1/p}\\ &\ll N^{1+\varepsilon}(q^{-1}+N^{-1}+qN^{-k})^{\omega_r}. \end{align*} $$

Since $\tau _p(k)=\omega _r$ in present circumstances, this completes the proof of the lemma.

Proof of Theorem 1.4.

Let $\mathscr A=\{a_1,a_2,\ldots \}$ , with $a_1<a_2<\ldots $ , be a strongly subconvex $L^p$ -set with $1\leqslant p<2$ . In order to facilitate tidier notation, we make a modest preliminary manoeuvre. Since $\mathscr A$ has positive lower density in $\mathbb N$ , we see that there exists a positive number $c=c(\mathscr A)$ having the property that when M is large, there exists an integer N with $N\leqslant cM$ for which one has $\{a_1,a_2,\ldots ,a_M\}=\mathscr A(N)$ . Next, let $k\geqslant 2$ , and define $\psi (n;\boldsymbol \alpha )$ according to (1.6). The hypotheses of the theorem permit us the assumption that one at least of the coefficients $\alpha _2,\ldots ,\alpha _k$ is irrational. We may therefore suppose that there is an integer h with $2\leqslant h\leqslant k$ for which $\alpha _k,\ldots ,\alpha _{h+1}$ are all rational, and $\alpha _h$ is irrational. We may suppose in addition that r is a natural number having the property that $r\alpha _j\in \mathbb Z\ (h+1\leqslant j\leqslant k)$ . In this context, if $h=k$ , then we set $r=1$ .

According to Weyl’s criterion, the sequence $(\psi (a_n;\boldsymbol \alpha ))_{n=1}^\infty $ is equidistributed modulo $1$ if and only if for any integer $m\ne 0$ , one has

$$\begin{align*}\lim_{M\rightarrow \infty}\frac{1}{M}\bigg| \sum_{\nu=1}^M e(m\psi(a_\nu;\boldsymbol \alpha))\bigg|=0. \end{align*}$$

The latter holds if and only if

$$\begin{align*}\lim_{N\rightarrow \infty}\frac{1}{N}\bigg| \sum_{n\in \mathscr A(N)}e(m\psi(n;\boldsymbol \alpha))\bigg|=0. \end{align*}$$

Thus, in the notation introduced in (4.1), our goal is to show that for any integer $m\ne 0$ , one has

(6.1) $$ \begin{align} |F_k(m\boldsymbol \alpha;N)|=o(N). \end{align} $$

Applying the notation (4.2) and (4.3) employed in the course of the proof of Theorem 1.2, we may infer from (4.4) that

$$\begin{align*}F_k(m\boldsymbol \alpha;N)=\int_0^1 G_k(m\boldsymbol \alpha,\beta)g(-\beta){\,\mathrm{d}}\beta. \end{align*}$$

An application of Hölder’s inequality reveals that

$$\begin{align*}|F_k(m\boldsymbol \alpha;N)|\leqslant \Big( \sup_{\beta\in [0,1)}|G_k(m\boldsymbol \alpha,\beta)|\Big)^{\frac{2}{p}-1} \Big( \int_0^1|G_k(m\boldsymbol \alpha,\beta)|^2{\,\mathrm{d}}\beta\Big)^{1-\frac{1}{p}}\Big( \int_0^1|g(\beta)|^p{\,\mathrm{d}}\beta \Big)^{\frac{1}{p}}. \end{align*}$$

Since $\mathscr A$ is a strongly subconvex $L^p$ -set with $1\leqslant p<2$ , we have

$$\begin{align*}\int_0^1|g(\beta)|^p{\,\mathrm{d}}\beta \ll N^{p-1}. \end{align*}$$

Then it follows via orthogonality that

(6.2) $$ \begin{align} F_k(m\boldsymbol \alpha;N)\ll (N)^{1-\frac{1}{p}}(N^{p-1})^{\frac{1}{p}}\Big( \sup_{\beta\in [0,1)} |G_k(m\boldsymbol \alpha,\beta)|\Big)^{\frac{2}{p}-1}. \end{align} $$

Thus, we may infer that the estimate (6.1) holds whenever one has

(6.3) $$ \begin{align} \sup_{{\beta}\in [0,1)}|G_k(m\boldsymbol \alpha,\beta)|=o(N). \end{align} $$

We establish (6.3) in a fairly routine manner via a sequence of intermediate steps. We begin by considering an auxiliary polynomial

$$\begin{align*}\omega(t;\boldsymbol \beta)=\beta_ht^h+\ldots +\beta_1t+\beta_0, \end{align*}$$

and the associated exponential sum

$$\begin{align*}U(\boldsymbol \beta ;N)=\sum_{1\leqslant n\leqslant N}e(\omega (n;\boldsymbol \beta)). \end{align*}$$

Suppose that $\beta _h$ is irrational. We claim that for any fixed value of $\eta>1$ , there exists a natural number $N_0$ having the property that

(6.4) $$ \begin{align} \sup_{\beta_0,\ldots ,\beta_{h-1}}|U(\boldsymbol \beta;N_0)|<N_0/\eta. \end{align} $$

We establish this claim by contradiction. Suppose, if possible, that for all large values of N, one has

$$\begin{align*}\sup_{\beta_0,\ldots ,\beta_{h-1}}|U(\boldsymbol \beta;N)|\geqslant N/\eta. \end{align*}$$

Then it follows from an appropriate version of Weyl’s inequality (see [Reference Baker2, Theorem 5.1]) that there exist $a\in \mathbb Z$ and $q\in \mathbb N$ with $(a,q)=1$ ,

$$\begin{align*}q<\eta^hN^{1/3}\quad \text{and}\quad |q\beta_h-a|<\eta^hN^{1/3-h}. \end{align*}$$

For each large natural number Q, we fix $N=\lfloor (Q/\eta ^h)^3\rfloor $ . Then we find that one has $q<Q$ and $|q\beta _h-a|<Q^{-3/2}$ . Since these inequalities hold for all large values of Q, it follows via a consideration of the continued fraction convergents to $\beta _h$ that this coefficient must be rational (compare [Reference Lê, Liu and Wooley19, Lemma 2.7]). This deduction contradicts our assumption that $\beta _h$ is irrational, and so we are forced to conclude that our claim (6.4) is indeed valid.

Now we return to consider the exponential sum $G_k(m\boldsymbol \alpha ,\beta )$ and the bound (6.3) that we seek to establish. Let $\eta>1$ be arbitrary. Put $\beta _h=mr^h\alpha _h$ . Then since one has $r\alpha _j\in \mathbb Z (h+1\leqslant j\leqslant k)$ , we see that

$$\begin{align*}m\psi(x+ry;\boldsymbol \alpha)+\beta (x+ry)\equiv \beta_hy^h+\ldots +\beta_1y+\beta_0\,\,(\mathrm{{mod}}\,\,1), \end{align*}$$

for some real numbers $\beta _0,\ldots ,\beta _{h-1}$ depending at most on r, m, x, $\beta $ and $\boldsymbol \alpha $ . Notice that since $\alpha _h$ is presumed to be irrational, so too is $\beta _h$ . Let $N_0$ be the natural number furnished in association with the upper bound (6.4) that we have established. When N is large enough in terms of r and $N_0$ , one has

$$\begin{align*}\bigg| \sum_{1\leqslant n\leqslant N}e(m\psi(n;\boldsymbol \alpha)+\beta n)-\frac{1}{N_0}\sum_{1\leqslant x\leqslant N} \sum_{1\leqslant y\leqslant N_0}e(m\psi(x+ry;\boldsymbol \alpha)+\beta (x+ry))\bigg| \leqslant 2rN_0. \end{align*}$$

Then we deduce from (6.4) that

$$ \begin{align*} \frac{1}{N}\sup_{\beta\in [0,1)}|G_k(m\boldsymbol \alpha,\beta)|&\leqslant \frac{1}{N_0}\sup_{\beta_0,\ldots ,\beta_{h-1}} |U(\boldsymbol \beta;N_0)|+\frac{2rN_0}{N}\\ &<\frac{1}{\eta}+\frac{2rN_0}{N}. \end{align*} $$

Therefore, for each integer $m\ne 0$ , we conclude that for every real number $\eta>1$ , one has

$$\begin{align*}\limsup_{N\rightarrow \infty}\frac{1}{N}\sup_{\beta\in [0,1)}|G_k(m\boldsymbol \alpha,\beta)|\leqslant \frac{1}{\eta}. \end{align*}$$

Since this upper bound holds for all $\eta>1$ , we infer that

$$\begin{align*}\lim_{N\rightarrow \infty}\frac{1}{N}\sup_{\beta\in [0,1)}|G_k(m\boldsymbol \alpha,\beta)|=0, \end{align*}$$

and the desired conclusion (6.3) follows. We therefore conclude that (6.1) holds, and so the sequence $(\psi (a_n;\boldsymbol \alpha ))_{n=1}^\infty $ is equidistributed modulo $1$ . This completes the proof of the theorem.

Proof. We may proceed just as in the proof of Theorem 1.4, though the hypothesis that $\mathscr A$ be weakly subconvex instead of strongly subconvex requires us to replace (6.2) by the bound

(6.5) $$ \begin{align} F_k(m\boldsymbol \alpha)\ll (N)^{1-\frac{1}{p}}(N^{p-1+\varepsilon})^{\frac{1}{p}} \Big( \sup_{\beta\in [0,1)}|G_k(m\boldsymbol \alpha ,\beta)|\Big)^{\frac{2}{p}-1}. \end{align} $$

Let $\varepsilon $ be a small positive number. Then it follows from Baker [Reference Baker2, Theorem 5.1] that whenever $P>N^{1-2^{-k}}$ and $|G_k(m\boldsymbol \alpha ,\beta )|>P$ , then there exist $q\in \mathbb N$ and $a_1,\ldots ,a_k\in \mathbb Z$ having the property that

$$\begin{align*}q<N^\varepsilon (NP^{-1})^k,\quad (q,a_1,\ldots ,a_k)=1,\quad (q,a_2,\ldots ,a_k)\leqslant N^\varepsilon, \end{align*}$$

$$\begin{align*}|qm\alpha_j-a_j|<(NP^{-1})^kN^{\varepsilon-j}\quad (2\leqslant j\leqslant k) \end{align*}$$

and

$$\begin{align*}|q(m\alpha_1+\beta)-a_1|<(NP^{-1})^kN^{\varepsilon-1}. \end{align*}$$

We may suppose that there is an index l with $2\leqslant l\leqslant k$ for which $\alpha _l$ is of finite Diophantine type. Thus, for some natural number r and positive number c, it follows that for all $a\in \mathbb Z$ one has

(6.6) $$ \begin{align} |qm\alpha_l-a|\geqslant c(qm)^{-r}. \end{align} $$

We take

$$\begin{align*}\varepsilon=\frac{1-p/2}{k2^kr}\quad \text{and}\quad P=N^{1-1/(k2^kr)}. \end{align*}$$

Then we deduce from the above discussion that whenever

$$\begin{align*}|G_k(m\boldsymbol \alpha ,\beta)|>N^{1-1/(k2^kr)}, \end{align*}$$

there exist $q\in \mathbb N$ and $a\in \mathbb Z$ with

$$\begin{align*}q<N^{2/(2^kr)}=(N^{1/r})^{2^{1-k}} \end{align*}$$

and

$$\begin{align*}|qm\alpha_l-a|<(N^{1/r})^{2^{1-k}}N^{-l}. \end{align*}$$

But then we find from (6.6) that

$$\begin{align*}c(qm)^{-r}\leqslant |qm\alpha_l-a|<(N^{1/r})^{2^{1-k}}N^{-l}\leqslant N^{-3/2}, \end{align*}$$

whence for large enough values of N, one sees that $q>N^{1/r}$ . Thus, we deduce that

$$\begin{align*}N^{1/r}<q<(N^{1/r})^{2^{1-k}}, \end{align*}$$

leading to a contradiction. We must therefore conclude that whenever N is sufficiently large in terms of m and $\alpha _l$ , one has

$$\begin{align*}|G_k(m\boldsymbol \alpha,\beta)|\leqslant N^{1-1/(k2^kr)}, \end{align*}$$

uniformly in $\beta $ . We thus deduce from (6.5) that

$$\begin{align*}F_k(m\boldsymbol \alpha)\ll N^{1+\frac{\varepsilon}{p}-\left(\frac{2}{p}-1\right) /(k2^kr)}= N^{1-\left(\frac{1}{p}-\frac{1}{2}\right) /(k2^kr)}, \end{align*}$$

whence, for each integer $m\ne 0$ , one has

$$\begin{align*}\lim_{N\rightarrow \infty}N^{-1}|F_k(m\boldsymbol \alpha)|=0. \end{align*}$$

This shows that

$$\begin{align*}\lim_{N\rightarrow \infty}\frac{1}{N}\bigg| \sum_{n=1}^Ne(m\psi(a_n;\boldsymbol \alpha))\bigg|=0, \end{align*}$$

so that $(\psi (a_n;\boldsymbol \alpha ))_{n=1}^\infty $ is equidistributed modulo $1$ . This completes the proof of the theorem.

Proof. Recalling our earlier notation (4.2), we see that

$$\begin{align*}\frac{1}{N}\sum_{n\in \mathscr A(N)}f(n)=\int_0^1 H_f(\alpha;N)g(-\alpha){\,\mathrm{d}}\alpha. \end{align*}$$

Hence, by Hölder’s inequality and Parseval’s identity, we find that

$$ \begin{align*} \frac{1}{N}\bigg| \sum_{n\in \mathscr A(N)}f(n)\bigg| &\leqslant \Big( \sup_{\alpha\in [0,1)}|H_f(\alpha;N)| \Big)^{\frac{2}{p}-1}\Big( \int_0^1 |H_f(\alpha;N)|^2{\,\mathrm{d}}\alpha \Big)^{1-\frac{1}{p}} \Big( \int_0^1|g(\alpha)|^p{\,\mathrm{d}}\alpha \Big)^{1/p}\\ &\ll \|H_f\|_\infty^{\frac{2}{p}-1}\left( N^{-1}\|f\|_{l^2(N)}^2\right)^{1-\frac{1}{p}}(N^{p-1})^{\frac{1}{p}}, \end{align*} $$

and the desired conclusion follows.

The proof of Theorem 1.5.

It is a consequence of work of Davenport [Reference Davenport10, equation (1)] that for any positive number A, one has

$$\begin{align*}\frac{1}{N}\sum_{1\leqslant n\leqslant N}\mu(n)e(n\theta)\ll \log^{-A}(2N), \end{align*}$$

uniformly in $\theta $ . Thus we have

$$\begin{align*}\|H_\mu\|_{\infty}\ll \log^{-A}(2N). \end{align*}$$

By a trivial estimate, moreover, one has

$$\begin{align*}\|\mu\|_{l^2(N)}=\Big( \frac{1}{N}\sum_{1\leqslant n\leqslant N}\mu(n)^2\Big)^{1/2}\leqslant 1. \end{align*}$$

Hence, we deduce from Theorem 7.1 that whenever $\mathscr A$ is a strongly subconvex $L^p$ -set for some $1\leqslant p<2$ , then

$$\begin{align*}\frac{1}{N}\sum_{n\in \mathscr A(N)}\mu(n)\ll \left( \log^{-A}(2N)\right)^{\frac{2}{p}-1}, \end{align*}$$

whence for any $B>0$ , one has

$$\begin{align*}\sum_{n\in \mathscr A(N)}\mu(n)\ll_B N\log^{-B}(2N). \end{align*}$$

This completes the proof of Theorem 1.5.

Proof. The hypotheses concerning $\theta $ ensure that the methods of Vinogradov and Hua (see [Reference Hua13, Reference Vinogradov25]) show that there is a positive number $\sigma =\sigma (k)$ having the property that

$$\begin{align*}\sum_{1\leqslant n\leqslant N}\Lambda (n)e(n^k\theta +\beta n)\ll N\log^{-\sigma A}(2N), \end{align*}$$

uniformly in $\beta $ . Thus, with $f(n)=\Lambda (n)e(n^k\theta )$ , one has $\|H_f\|_\infty \ll \log ^{-\sigma A}(2N)$ and

$$\begin{align*}\|f\|_{l^2(N)}\leqslant \bigg( \frac{1}{N}\sum_{1\leqslant n\leqslant N}\Lambda(n)^2\bigg)^{1/2}\ll (\log (2N))^{1/2}. \end{align*}$$

Hence, we deduce from Theorem 7.1 that

$$\begin{align*}\sum_{n\in \mathscr A(N)}\Lambda(n)e(n^k\theta)\ll N\left( \log^{-\sigma A}(2N)\right)^{\frac{2}{p}-1} (\log (2N))^{1-\frac{1}{p}}. \end{align*}$$

The desired conclusion now follows.

Proof. We apply Theorem 7.1 with $f(n)=a(n)$ . Here, as a consequence of Iwaniec [Reference Iwaniec15, Theorem 5.3], one has

$$\begin{align*}\|H_f\|_\infty =\sup_{\beta\in [0,1)}\bigg| \frac{1}{N}\sum_{1\leqslant n\leqslant N}a(n)e(\beta n)\bigg| \ll N^{-1+k/2}\log (2N), \end{align*}$$

whilst [Reference Iwaniec15, Theorem 5.1] provides the bound

$$\begin{align*}\|f\|_{l^2(N)}=\bigg( \frac{1}{N}\sum_{1\leqslant n\leqslant N}|a(n)|^2\bigg)^{1/2}\ll N^{(k-1)/2}. \end{align*}$$

Consequently, we see from Theorem 7.1 that

$$ \begin{align*} \sum_{n\in \mathscr A(N)}a(n)&\ll N\left( N^{\frac{k}{2}-1}\log (2N)\right)^{\frac{2}{p}-1} \left( N^{\frac{k-1}{2}}\right)^{2-\frac{2}{p}}\\ &\ll N^{\frac{k}{2}+1-\frac{1}{p}}(\log (2N))^{\frac{2}{p}-1}. \end{align*} $$

This completes the proof of the corollary.

Proof. We apply Theorem 7.1 with $f(n)=\chi (n)$ . Here, as a consequence of Chamizo [Reference Chamizo9, Theorem 1.1], one has

$$\begin{align*}\| H_f\|_\infty = \sup_{\beta\in [0,1)}\bigg| \frac{1}{N}\sum_{1\leqslant n\leqslant N}\chi(n)e(\beta n)\bigg| \ll N^{-\frac{1}{r}}\pi^{\frac{1}{4r-4}}(\log \pi)^{\frac{1}{2r}}, \end{align*}$$

whilst $\|f\|_{l^2(N)}\leqslant 1$ . Consequently, we find that the desired conclusion is immediate from Theorem 7.1.

Proof. We apply Theorem 7.1 with $f(n)=t(n)$ . In this setting, as a consequence of Epstein, Hafner and Sarnak [Reference Epstein, Hafner and Sarnak11, Theorem 3.1], one has

$$\begin{align*}\| H_f\|_\infty = \sup_{\beta\in [0,1)}\bigg| \frac{1}{N}\sum_{1\leqslant n\leqslant N}t(n)e(\beta n)\bigg| \ll N^{\varepsilon-1/2}, \end{align*}$$

uniformly in $\beta $ , whilst [Reference Epstein, Hafner and Sarnak11, equation (2.2)] confirms that $\|f\|_{l^2(N)}\ll 1$ . Consequently, the desired conclusion is immediate from Theorem 7.1.

The proof of Theorem 1.6.

We adjust the notation (3.2) by writing

(8.2) $$ \begin{align} f_N(\alpha)=\sum_{n\in \mathscr A(N)}e(n\alpha). \end{align} $$

Thus, when $1<p<2$ , one has

(8.3) $$ \begin{align} |f_N(\alpha)|^p=f_N(\alpha)^{p/2}f_N(-\alpha )^{p/2}. \end{align} $$

Equipped with this relation, we seek to convert the discrete mean value $S_p(N;\mathscr A;q)$ into a continuous analogue by applying the Sobolev-Gallagher inequality (see [Reference Montgomery21, Lemma 1.1]). Thus, whenever $a\in \mathbb Z$ and $q\in \mathbb N$ , we have

$$\begin{align*}|f_N(a/q)|^p\leqslant q\int_{-1/(2q)}^{1/(2q)}|f_N(\beta+a/q)|^p{\,\mathrm{d}}\beta + \frac{1}{2}\int_{-1/(2q)}^{1/(2q)}\Big| \frac{\mathrm{d}\,}{\mathrm{d}\beta} |f_N(\beta+a/q)|^p\Big| {\,\mathrm{d}}\beta. \end{align*}$$

When $f_N(\beta +a/q)\ne 0$ , it follows from (8.3) that

$$\begin{align*}\frac{\mathrm{d}\,}{\mathrm{d}\beta} |f_N(\beta+a/q)|^p=\frac{p}{2}|f_N(\beta+a/q)|^p \bigg( \frac{f_N'(\beta+a/q)}{f_N(\beta+a/q)}-\frac{f_N'(-\beta-a/q)}{f_N(-\beta-a/q)}\bigg) , \end{align*}$$

whence

$$\begin{align*}\Big| \frac{\mathrm{d}\,}{\mathrm{d}\beta} |f_N(\beta+a/q)|^p\Big| \leqslant p|f_N(\beta+a/q)|^{p-1}|f^{\prime}_N(\beta+a/q)|. \end{align*}$$

Since $p>1$ , we therefore deduce from (8.1) that

(8.4) $$ \begin{align} S_p(N;\mathscr A;q)\leqslant \int_0^1|f_N(\alpha)|^p{\,\mathrm{d}}\alpha + \frac{p}{2q}\int_0^1|f_N(\alpha)|^{p-1}|f^{\prime}_N(\alpha)|{\,\mathrm{d}}\alpha. \end{align} $$

We must now bound mean values involving $f^{\prime }_N(\alpha )$ in terms of mean values involving $f_N(\alpha )$ . For this purpose, we draw inspiration from the argument following [Reference Wooley27, equation (3.2)]. We observe that

$$\begin{align*}f_N'(\alpha)=2\pi \mathrm{i}\sum_{n\in \mathscr A(N)}ne(n\alpha). \end{align*}$$

Recall the definition (8.1). Then, on writing $\mathscr A(N)=\{ a_1,a_2,\ldots ,a_M\}$ , with

$$\begin{align*}a_1<a_2<\ldots <a_M, \end{align*}$$

we see that the sum on the right hand side here may be rewritten via partial summation in the form

$$ \begin{align*} \sum_{m=1}^Ma_me(a_m\alpha)&=a_1f_{a_1}(\alpha)+ \sum_{m=2}^Ma_m\left( f_{a_m}(\alpha)-f_{a_{m-1}}(\alpha)\right) \\ &=a_Mf_{a_M}(\alpha)-\sum_{m=1}^{M-1}(a_{m+1}-a_m)f_{a_m}(\alpha). \end{align*} $$

We therefore see that for all real numbers $\alpha $ , one has

$$ \begin{align*} \bigg| \sum_{m=1}^Ma_me(a_m\alpha)\bigg|&\leqslant a_M|f_{a_M}(\alpha)|+\left( \max_{1\leqslant w\leqslant M-1}|f_{a_w}(\alpha)| \right) \sum_{1\leqslant m\leqslant M-1}|a_{m+1}-a_m|\\ &\leqslant (a_M+(a_M-a_1))\max_{1\leqslant w\leqslant M}|f_{a_m}(\alpha)|\\ &\leqslant 2N\max_{1\leqslant n\leqslant N}|f_n(\alpha)|. \end{align*} $$

An application of the Carleson-Hunt theorem (see [Reference Hunt14, Theorem 1]) in the form given by Bombieri [Reference Bombieri5, page 12] shows that

$$\begin{align*}\int_0^1\left( \max_{1\leqslant n\leqslant N}|f_n(\alpha)|\right)^p{\,\mathrm{d}}\alpha \ll_p \int_0^1|f_N(\alpha)|^p{\,\mathrm{d}}\alpha. \end{align*}$$

We therefore deduce that whenever $\mathscr A$ is a strongly subconvex $L^p$ -set, then one has

(8.5) $$ \begin{align} \int_0^1|f^{\prime}_N(\alpha)|^p{\,\mathrm{d}}\alpha &\ll N^p\int_0^1\left( \max_{1\leqslant n\leqslant N}|f_n(\alpha)|\right)^p{\,\mathrm{d}}\alpha \notag \\ &\ll N^p\int_0^1|f_N(\alpha)|^p{\,\mathrm{d}}\alpha \notag \\ &\ll N^{2p-1}. \end{align} $$

We now substitute the estimate (8.5) into (8.4) following an application of Hölder’s inequality. Thus, on applying once again the hypothesis that $\mathscr A$ be a strongly subconvex $L^p$ -set, we find that

$$ \begin{align*} S_p(N;\mathscr A;q)&\ll N^{p-1}+\frac{1}{q}\bigg( \int_0^1|f_N(\alpha)|^p{\,\mathrm{d}}\alpha\bigg)^{1-1/p} \bigg( \int_0^1|f_N'(\alpha)|^p{\,\mathrm{d}}\alpha \bigg)^{1/p}\\ &\ll N^{p-1}+\frac{1}{q}(N^{p-1})^{1-1/p}(N^{2p-1})^{1/p}\\ &\ll N^{p-1}+N^p/q. \end{align*} $$

This delivers the conclusion of the theorem.

Proof. We recall the notation (8.2), and in addition define

$$\begin{align*}T(a)=\sum_{m=1}^q\chi(m)e(ma/q). \end{align*}$$

Then, by orthogonality, one sees that

$$\begin{align*}\sum_{n\in \mathscr A(N)}\chi(n)=\frac{1}{q}\sum_{a=1}^qT(a)f_N(-a/q). \end{align*}$$

An application of Hölder’s inequality reveals, therefore, that

(8.6) $$ \begin{align} \bigg| \sum_{n\in \mathscr A(N)}\chi(n)\bigg| \leqslant \bigg( \frac{1}{q}\sum_{a=1}^q|f_N(a/q)|^p\bigg)^{1/p} \bigg( \frac{1}{q}\sum_{a=1}^q|T(a)|^{p/(p-1)}\bigg)^{1-1/p}. \end{align} $$

Since we assume $\chi $ to be primitive, it follows from [Reference Montgomery and Vaughan22, Theorem 9.7] that $|T(a)|\leqslant \sqrt {q}$ . Thus, by orthogonality, we see that whenever $1<p<2$ , one has

$$\begin{align*}\frac{1}{q}\sum_{a=1}^q|T(a)|^{\frac{p}{p-1}}\leqslant (\sqrt{q})^{\frac{2-p}{p-1}}\frac{1}{q}\sum_{a=1}^q|T(a)|^2 \leqslant q^{\frac{p}{2p-2}}. \end{align*}$$

We therefore deduce from Theorem 1.6 and (8.6) that

$$ \begin{align*} \bigg| \sum_{n\in \mathscr A(N)}\chi(n)\bigg| &\ll \left( N^{p-1}(1+N/q)\right)^{1/p} \left( q^{p/(2p-2)}\right)^{1-1/p}\\ &\ll q^{1/2}N^{1-1/p}(1+N/q)^{1/p}. \end{align*} $$

This yields the conclusion of the corollary.