2.1. Useful ingredients

We will make repeated use of the Siegel–Walfisz theorem [Reference Hua10, lemma 7.14], which we now state for convenience. Recall that the logarithmic integral is given by

\begin{equation*} {\mathrm{Li}}(x) = \int_2^x \frac{{\,{\rm d}} t}{\log t} \qquad (x \geqslant 2). \end{equation*}

Theorem 2.1 (Siegel–Walfisz theorem)

Let $P \geqslant 2$, and write $L = \log P$. Let A > 0. Then, there exists $c = c(A) \gt 0$ such that the following is true. Let $n,q,a \in \mathbb{Z}$ with

\begin{equation*} 1 \leqslant n \leqslant P, \qquad 1 \leqslant q \leqslant L^A. \end{equation*}

Then,

\begin{equation*} \# \{p \leqslant n: p \equiv a \;(\operatorname{mod}{q}) \} = \frac{{\mathrm{Li}}(n)}{\varphi(q)} + O(Pe^{-c \sqrt{\log L}}). \end{equation*}

We will also make repeated use of [Reference Rice19, lemma 9], which we state below. Owing to an inaccuracy in the published version of Rice’s article, we cite a later arXiv version. This is also explained in a remark immediately following [Reference Rice19, lemma 9]. The fact that C only depends on the degree comes from the proof.

Lemma 2.2. For any integer $k\geqslant 2$, there exists $C = C(k) \gt 0$ such that the following holds. Let $g(x) = a_k x^k + \cdots + a_1 x + a_0 \in \mathbb{Z}[x]$, and let $W, b \in \mathbb{Z}$. Let $a \in \mathbb{Z}$ and $q \in \mathbb{N}$ be coprime. Let ${\omega}(q)$ denote the number of distinct prime factors of q. Let $q = q_1 q_2$, where q ₂ is the greatest divisor of q that is coprime to W, and let ${\mathrm{cont}}(g) = \gcd(a_1,\ldots,a_k)$. Then,

\begin{equation*} \left| \sum_{\substack{\ell=0\\(W\ell+b,q)=1}}^{q-1} e_q(ag(\ell)) \right| \leqslant C^{{\omega}(q)} \left( \gcd({\mathrm{cont}}(g),q_1) \gcd(a_k,q_2) \right)^{1/k} q^{1-1/k}. \end{equation*}

We note from its proof that the general epsilon-removal lemma [Reference Salmensuu21, lemma 25] holds with $\| {\alpha} - a/q \|^{\kappa}$ in place of $\| {\alpha} - a/q \|$, and for complex-valued f, with the notation therein. For convenience, we state this version below.

Lemma 2.3. (Epsilon-removal lemma)

Let ${\kappa}, \epsilon \gt 0$, $N \in \mathbb{N}$, $K \geqslant 1$, $u \gt 2/{\kappa}$, and $v \gt u + \epsilon$. Let $\phi: [N] \to [0,\infty)$ and $f: [N] \to \mathbb{C}$ with

\begin{equation*} |f(n)| \leqslant \phi(n) \qquad (n \in [N]). \end{equation*}

Let C be a large, positive constant, and let

\begin{equation*} Q \gt C + K^{1 + 2/({\kappa} \epsilon)}, \qquad T \gt 2Q^2. \end{equation*}

Let $\mathfrak M$ be the union of the sets

\begin{equation*} \mathfrak M(q,a) = \{{\alpha} \in [0,1]: |{\alpha} - a/q| \leqslant 1/T \} \end{equation*}

over coprime integers $0 \leqslant a \leqslant q \leqslant Q$. Assume that

\begin{equation*} \sum_{n \leqslant N} \phi(n) \ll N, \qquad \| \hat f \|_u^u \ll K N^{u-1}. \end{equation*}

Assume, further, that

\begin{equation*} \hat \phi({\alpha}) \ll \frac{q^{-{\kappa}}N}{1 + N \| {\alpha} - a/q \|^{\kappa}} + o(K^{-2/\epsilon}N) \qquad ({\alpha} \in \mathfrak M) \end{equation*}

and

\begin{equation*} \hat \phi({\alpha}) = o(K^{-2/\epsilon}N) \qquad ({\alpha} \notin \mathfrak M). \end{equation*}

Then,

\begin{equation*} \| \hat f \|_v^v \ll_v N^{v-1}. \end{equation*}

Here, $o(K^{-2/\epsilon}N)$ denotes any quantity X such that if c > 0 and N is sufficiently large, then

\begin{equation*} X \leqslant c K^{-2/\epsilon} N. \end{equation*}

This quantity may differ between instances.

2.2. Necessary conditions

We now provide necessary conditions for Eq. (1.3) to be partition or density regular over the primes. To state our results, we recall that an integer polynomial h is called intersective (or intersective of the first kind) if, for each $n\in\mathbb{N}$, there exists $x\in\mathbb{Z}$ such that $h(x)\equiv 0\;(\operatorname{mod}{n})$. We call h intersective of the second kind if this statement holds under the additional condition such that an x can be found, which is coprime to n. The following lemma demonstrates that intersectivity is a necessary condition for partition regularity of general polynomial equations.

Lemma 2.4. Let $s\in\mathbb{N}$ and let $F\in\mathbb{Z}[x_1,\ldots,x_s]$. Consider the equation

(2.1)\begin{equation} F(x_1,\ldots,x_s) = 0. \end{equation}

1. (PR) If (2.1) is partition regular (over the primes), then the single-variable polynomial $F(x,\ldots,x)\in\mathbb{Z}[x]$ is intersective (of the second kind).

2. (DR) If (2.1) is density regular or density regular over the primes, then $F(x,\ldots,x)$ is the zero polynomial.

We now apply this lemma to (1.3) to establish the ‘only if’ parts of theorem 1.1. By working modulo $|\mu| n$ for any $n \in \mathbb{N}$, we see that if $\mu \ne 0$ and $\mu h$ is intersective of the second kind, then so too is h. Note also that the following result does not impose any restriction on the number of variables:

Proof. Throughout this proof, we write $\mu =a_1+\cdots+a_s$ and $H(x) =\mu h(x) - b$. First suppose (1.3) is partition regular over the primes. Applying lemma 2.4 to the polynomial $F(x_1,\ldots,x_s) = a_1x_1 + \cdots + a_sx_s - b$, we find that H(x) is intersective of the second kind. By considering solutions to $H(x)\equiv 0\;(\operatorname{mod}{d})$ for any $d\mid \mu$, we observe that $\mu\mid b$. If µ ≠ 0, then there is a unique $m\in\mathbb{Z}$ with $b=\mu m$ such that $H(x)=\mu(h(x) - m)$, whence $h(x)-m$ is intersective of the second kind. If µ = 0, then b = 0, and so, upon taking $m=h(1)$, we have $b=\mu m$, and $h(x)-m$ is trivially intersective of the second kind. In both cases, Eq. (1.3) becomes

\begin{equation*} \sum_{i=1}^{s} a_i(h(x_i) - m) = 0, \end{equation*}

for some $m\in\mathbb{Z}$ such that $h(x)-m$ is intersective of the second kind. As this new equation is partition regular, we infer from [Reference Chapman and Chow2, proposition 2.1] the existence of a set $I\subseteq\{1,\ldots,s\}$ with the desired properties.

Finally, suppose that (1.3) is density regular or density regular over the primes. Then, lemma 2.4 implies that $H(x) = \mu h(x) - b$ is the zero polynomial. Since h has positive degree, we conclude that $b=\mu=0$.

In view of these necessary conditions, theorem 1.1 is now an immediate consequence of theorem 1.5.

2.3. Linear form equations

Having dispensed with the necessary conditions for partition and density regularity, we focus on finding monochromatic or dense solutions to (1.3). The necessary conditions we have established, therefore, inform us that (1.3) takes the shape

\begin{equation*} \sum_{i\in I}a_i (h(x_i)-m) = -\sum_{j\in[s]\setminus I}a_j (h(x_j)-m), \end{equation*}

where $I\subseteq [s]$ is non-empty with $\sum_{i\in I}a_i=0$, and $m\in\mathbb{Z}$ is such that $b=(a_1+\cdots+a_s)m$ and $h(x)-m$ is intersective of the second kind. Upon replacing h(x) with $h(x)-m$, we can therefore reduce to the case where b = 0 and h(x) is intersective of the second kind.

To find monochromatic or dense solutions to (1.3) with b = 0, we study equations of the form

(2.2)\begin{equation} L_1(h(\mathbf{x})) = L_2(h(\mathbf{y})), \end{equation}

for some linear forms L ₁ and L ₂. To avoid trivialities, we only consider non-degenerate linear forms, where $L(\mathbf{x})=a_1x_1 + \cdots + a_sx_s$ is non-degenerate if $a_i\neq 0$ for all $i\in[s]$. For this new equation, the necessary conditions for partition and density regularity become $L_1(1,\ldots,1)=0$. Following the recent works [Reference Chapman and Chow2, Reference Chow, Lindqvist and Prendiville5], we address both density and partition regularity for (2.2) simultaneously by seeking solutions where the x_i variables are sourced in a dense subset of $\mathcal P_X$, while the remaining y_j variables come from a colour class $\mathcal C_k\subseteq \mathcal P_X$.

Before proceeding to our results, we require some notation. We begin by providing an explicit description of the threshold $s_0(d)$ for the number of variables required in our main theorems. Let $T = T(d) \in \mathbb{N}$ be minimal such that if $h(x) \in \mathbb{Z}[x]$ has degree d, then

\begin{equation*} h(x_1) + \cdots + h(x_T) = h(x_{T+1}) + \cdots + h(x_{2T}) \end{equation*}

has $O_{h,\varepsilon}(X^{2T - d + \varepsilon})$ solutions $\mathbf{x} \in [X]^{2T}$. Equivalently, by orthogonality, $T = T(d)$ is the smallest positive integer such that the moment estimate

\begin{equation*} \int_{\mathbb{T}}\left\lvert \sum_{x\leqslant X}e(\alpha h(x))\right\rvert^{2T}\ll_{h,\varepsilon} X^{2T-d+\varepsilon} \end{equation*}

holds for any integer polynomial h of degree d. The quantity $s_0(d)$ appearing in theorem 1.1 is now defined to be $s_0(d):= 2T(d) + 1$.

It follows from Hua’s lemma [Reference Hua9, equation (1)] that $T(2) \leqslant 2$ and $T(3) \leqslant 4$. In general, the proof of [Reference Wooley23, corollary 14.7] delivers

\begin{equation*} T(d) \leqslant \frac{d(d-1)}2 + \lfloor \sqrt{2d+2} \rfloor. \end{equation*}

These observations verify the bound (1.4) for $s_0(d)$. Finally, by considering solutions with $x_{i}=x_{i+T}$ for $i=1,2,\ldots,T$, we record the lower bounds

(2.3)\begin{equation} T(d) \geqslant d, \qquad s_0(d) \geqslant 2d + 1. \end{equation}

We can now state our main result on partition and density regularity over primes for linear form equations (2.2).

Theorem 2.6 Let r and $d\geqslant 2$ be positive integers, and let $0 \lt \delta\leqslant 1$. Let h be an integer polynomial of degree d, which is intersective of the second kind. Let $s \geqslant 1$ and $t \geqslant 0$ be integers such that $s + t \geqslant s_0(d)$. Let

\begin{equation*} L_1(\mathbf{x}) \in \mathbb{Z}[x_1,\ldots,x_s], \qquad L_2(\mathbf{y}) \in \mathbb{Z}[y_1,\ldots,y_t] \end{equation*}

be non-degenerate linear forms such that $L_1(1,\ldots,1) = 0$. Then, there exist

\begin{equation*} X_0=X_0(\delta,h,r,L_1,L_2)\in\mathbb{N}, \qquad \tau_0(\delta)=\tau_0(h,r,L_1,L_2;\delta)\in (0,1) \end{equation*}

such that the following is true for all $X\geqslant X_0$. Suppose $\mathcal P_X = \mathcal C_1 \cup \cdots \cup \mathcal C_r$. Then, there exists $k \in [r]$ with $|\mathcal C_k|\geqslant\tau_0(\delta) |\mathcal P_X|$ such that if $A \subseteq \mathcal P_X$ satisfies $|A| \geqslant {\delta} |\mathcal P_X|$, then

\begin{equation*} \# \{(\mathbf{x}, \mathbf{y}) \in A^s \times \mathcal C_k^t: L_1(h(\mathbf{x})) = L_2(h(\mathbf{y})) \} \gg \frac{X^{s+t-d}}{(\log X)^{s+t}}. \end{equation*}

The implied constant may depend on $h, L_1, L_2, r, \delta$.

Remark 2.7. In the case t = 0, we have a linear form L ₂ in zero variables, and we are counting solutions $\mathbf{x}\in A^s$ to the equation

\begin{equation*} L_1(h(\mathbf{x})) = 0. \end{equation*}

Note that when t = 0, all linear forms L ₂ in t variables are vacuously non-degenerate.

By harnessing a combinatorial ‘cleaving’ argument of Prendiville [Reference Prendiville17], we can swiftly deduce theorem 1.5 from theorem 2.6.

Proof of theorem 1.5 given theorem 2.6

Following the argument given at the beginning of this subsection, we may reduce to the case where b = 0 and h is intersective of the second kind. Combining [Reference Chapman and Chow2, lemma 3.2] with (2.3), for N sufficiently large, the number of solutions $\mathbf{x} \in [N]^s$ to (1.3) such that $x_i = x_j$ holds for some i ≠ j is $O_\varepsilon(N^{s-d+\varepsilon-1/2})$. Therefore, by setting t = 0, we see that the density regularity statement in theorem 1.5 follows directly from theorem 2.6. Similarly, given a colouring $\mathcal P_N = \mathcal C_1\cup\cdots\cup\mathcal C_r$, provided N is sufficiently large, it remains to show that there are at least $c_1N^{-d}(N/\log N)^{s}$ monochromatic solutions to (2.2) with

\begin{equation*} L_1(\mathbf{x}):=\sum_{i\in I}a_ix_i, \quad\text{and} \quad L_2(\mathbf{y}):= -\sum_{i\in [s] \setminus I}a_iy_j. \end{equation*}

For each δ > 0, let $\tau_0(\delta)\in(0,1)$ be as given in the statement of theorem 2.6. By making minor adjustments, we may assume that $\tau_0(\delta)\leqslant\delta$. Set $\delta_0=1/r$, and for each $i\in[r]$, let $\delta_i:=\tau_0(\delta_{i-1})$, whence $0 \lt \delta_r\leqslant\ldots\leqslant\delta_0 \lt 1$. Take $N\geqslant X_0(\delta_r,h,r,L_1,L_2)$ as in theorem 2.6, and suppose $\mathcal P_N =\mathcal C_1\cup\cdots\cup\mathcal C_r$. For each $0\leqslant i\leqslant r$, let $k_i\in[r]$ be the index given by applying theorem 2.6 with $\delta=\delta_i$. By the pigeonhole principle, we can find $k\in[r]$ and $0\leqslant i \lt j\leqslant r$ such that $k_i=k_j=k$. Therefore,

\begin{equation*} \frac{|\mathcal C_k|}{|\mathcal P_N|} \geqslant \tau_0(\delta_i) =\delta_{i+1}\geqslant\delta_j, \end{equation*}

and

\begin{equation*} \# \{(\mathbf{x}, \mathbf{y}) \in A^{|I|} \times \mathcal C_k^{s-|I|}: L_1(h(\mathbf{x})) = L_2(h(\mathbf{y})) \} \gg_{\delta_j,h,r,L_1,L_2} \frac{N^{s-d}}{(\log N)^{s}} \end{equation*}

holds for any $A\subseteq\mathcal P_N$ with $|A|\geqslant\delta_j|\mathcal P_N|$. Taking $A=\mathcal C_k$ finishes the proof.

2.4. Auxiliary intersective polynomials

The next step of our argument is to use a version of Green’s Fourier-analytic transference principle [Reference Green6] to obtain solutions to (2.2) by ‘transferring’ solutions from a ‘linearized’ equation. To make this precise, we first need to introduce the auxiliary intersective polynomials of Lucier [Reference Lucier15], which emerge during the execution of this process.

Let h be an integer polynomial of positive degree d, which is intersective of the second kind. Thus, for each prime p, we can find a p-adic unit $z_p \in \mathbb{Z}_p^{\times}$ such that $h(z_p)=0$. Throughout this article, we fix a choice of z_p for each prime p and let m_p be the multiplicity of z_p as a zero of h. For each prime p and positive integer D, let $\mathrm{ord}_p(D)$ denote the largest non-negative integer n such that pⁿ divides D. We can then define the completely multiplicative function

(2.4)\begin{equation} {\lambda}(D) := \prod_p p^{m_p \mathrm{ord}_p(D)} \qquad (D\in\mathbb{N}). \end{equation}

By noting that $1\leqslant m_p\leqslant \mathrm{deg}(h)=d$ for all p, we have

(2.5)\begin{equation} D \mid {\lambda}(D) \mid D^d. \end{equation}

For each prime p and non-negative integer k, reducing z_p modulo p^k reveals that there exists a unique residue $x\in\mathbb{Z}/p^k\mathbb{Z}$ such that $x\equiv z_p \;(\operatorname{mod}{p^k\mathbb{Z}_p})$. By the Chinese remainder theorem, we can therefore find a unique integer r_D in the range $(-D,0]$, which satisfies

\begin{equation*} r_D \equiv z_p \;(\operatorname{mod}{p^{\mathrm{ord}_p(D)} \mathbb{Z}_p}) \end{equation*}

for all primes p. As h is intersective of the second kind, we have $(r_D, D) = 1$.

Finally, with this notation in place, we define the auxiliary intersective polynomial

\begin{equation*} h_D(x) := \frac{h(r_D + Dx)}{{\lambda}(D)} \in \mathbb{Z}[x]. \end{equation*}

These polynomials and the surrounding notation were introduced by Lucier [Reference Lucier15], who also showed that h_D is indeed a polynomial with integer coefficients [Reference Lucier15, lemma 21]. The most important property of these auxiliary polynomials is that the greatest common divisor of the non-constant coefficients of h_D is bounded uniformly in D. Specifically, for all $D\in\mathbb{N}$, [Reference Lucier15, lemma 28] states that

\begin{equation*} \gcd(h_D - h_D(0)) \ll_h 1. \end{equation*}

As in [Reference Chapman and Chow2, §6], this bound is crucial to our investigation of exponential sums involving intersective polynomials (see §7 and §9).

We can now state our linearized version of theorem 2.6.

Theorem 2.8 Let r and $d\geqslant 2$ be positive integers, and let $0 \lt \delta\leqslant 1$. Let h be an integer polynomial of degree d which is intersective of the second kind. Let $s \geqslant 1$ and $t \geqslant 0$ be integers such that $s + t \geqslant s_0(d)$. Let

\begin{equation*} L_1(\mathbf{x}) \in \mathbb{Z}[x_1,\ldots,x_s], \qquad L_2(\mathbf{y}) \in \mathbb{Z}[y_1,\ldots,y_t] \end{equation*}

be non-degenerate linear forms such that $L_1(1,\ldots,1) = 0$. Then, there exists

\begin{equation*} Z_0=Z_0(D, h, r, {\delta}, L_1, L_2)\in\mathbb{N}\quad \text{and}\quad\eta = \eta(d,\delta,L_1,L_2) \in (0,1) \end{equation*}

such that the following is true. Let $D, Z \in \mathbb{N}$ satisfy $Z \geqslant Z_0$, and set $N:=h_D(Z)$. Suppose

\begin{equation*} [\eta Z,Z]\cap\{z \in [Z]: r_D + Dz \in \mathcal P \} = \mathcal C_1 \cup \cdots \cup \mathcal C_r. \end{equation*}

Then, there exists $k \in [r]$ such that if $\mathcal A\subseteq[N]$ satisfies $|\mathcal A|\geqslant\delta N$, then

\begin{equation*} \# \{({\mathbf{n}},\mathbf{z}) \in \mathcal A^s \times \mathcal C_k^t: L_1({\mathbf{n}}) = L_2(h_D(\mathbf{z})) \} \gg N^{s-1} \left(\frac{DZ}{\varphi(D)\log Z} \right)^t. \end{equation*}

The implied constant may depend on $h, L_1, L_2, r, \delta$.

The proof of theorem 2.8 is deferred to the final two sections of this article. As in [Reference Chapman and Chow2], we prove this ‘linearized’ result by applying an arithmetic regularity lemma. To streamline this forthcoming argument, we require the following proposition, which is a minor variation of [Reference Chapman and Chow2, proposition 3.10] and is proved in the same way.

2.5. Sketch of the transference argument

In this subsection, we outline how the transference principle allows us to deduce theorem 2.6 from theorem 2.8. Fix an integer polynomial h that is intersective of the second kind, as well as a pair of linear forms L ₁ and L ₂ as in the statement of theorem 2.6. We begin by recalling that theorem 2.6 concerns the equation

(2.6)\begin{equation} L_1(h(\mathbf{x})) = L_2(h(\mathbf{y})), \end{equation}

while theorem 2.8 considers, for some parameter $D\in\mathbb{N}$, the ‘linearized’ equation

(2.7)\begin{equation} L_1({\mathbf{n}}) = L_2(h_D(\mathbf{z})). \end{equation}

Suppose we have a finite colouring $\mathcal P_X= \mathcal C_1\cup\cdots\cup \mathcal C_r$ and a set $A\subseteq\mathcal P_X$ with $|A|\geqslant\delta|\mathcal P_X|$. For the convenience of this sketch, assume that $X\equiv r_D \;(\operatorname{mod}{D})$. Choosing $Z\in\mathbb{N}$ such that $X=r_D + DZ$, we can define an r-colouring

\begin{equation*} \{z \in [Z]: r_D + Dz \in \mathcal P \} = \tilde{\mathcal C}_1 \cup \cdots \cup \tilde{\mathcal C}_r \end{equation*}

by

\begin{equation*} \tilde{\mathcal C_i}:=\{z\in [Z] : r_D + Dz\in\mathcal C_i\}. \end{equation*}

Let $N:=h_D(Z)$. By pigeonholing, we find a ‘dense’ set $\mathcal A\subseteq[N]$ such that

\begin{equation*} \mathcal A\subseteq \left\{\frac{h(x)-h(b)}{\lambda(D)}: x \in A \right \}, \end{equation*}

for some integer b.

Theorem 2.8 now informs us that there are many solutions $({\mathbf{n}},\mathbf{z})\in \mathcal A^s\times\tilde{\mathcal C}_k^t$ to (2.7) for some $k\in[r]$. Given such a solution, our construction of $\mathcal A$ and $\tilde{\mathcal C}_k$ furnishes a solution $(\mathbf{x},\mathbf{y})\in A^s\times\mathcal C_k^t$ to (2.6) satisfying

\begin{equation*} n_i = \frac{h(x_i) - h(b)}{\lambda(D)}, \quad y_j = r_D + Dz_j \qquad (1\leqslant i\leqslant s, \quad 1\leqslant j\leqslant t). \end{equation*}

Since the map $({\mathbf{n}},\mathbf{z})\mapsto(\mathbf{x},\mathbf{y})$ is injective, this argument allows us to obtain many solutions to (2.6). However, observe that the number of solutions to (2.6) given in theorem 2.6 is $ X^{s-d+t+o(1)}, $ which is far fewer than the number of solutions to (2.7) provided by theorem 2.8, namely $X^{ds - d + t + o(1)}$. This shortfall is handled by instead considering weighted counts of solutions to (2.6). Our task is then to construct an appropriate weight function ν, which is supported on the set

\begin{equation*} [N]\cap\left\{\frac{h(x)-h(b)}{\lambda(D)}: b \lt x \leqslant X\right \}. \end{equation*}

The key utility of the transference principle is that, provided our weight function is suitably ‘pseudorandom’, we can find a ‘dense model’ $g:[N]\to[0,1]$ such that $\hat{\nu}\approx\hat{g}$. Applying theorem 2.8 to a set of the form $\mathcal A=\{x\in[N]: g(x) \gt c\}$, our argument above allows us to prove theorem 2.6.

To ensure our weight ν is sufficiently pseudorandom, we have to contend with the fact that the set $h(\mathcal P)$ is not equidistributed in residue classes. This issue prevents one from simply taking ν to be a scaled version of the indicator function of $h(\mathcal P)$. Fortunately, there is a standard technical manoeuvre, known as the W-trick, developed by Green [Reference Green6] to account for equidistribution modulo small primes. In the setting discussed above, this amounts to demanding that our weight ν is supported on a set of the form

\begin{equation*} [N]\cap\left\{\frac{h(x)-h(b)}{\lambda(D)}: b \lt x \leqslant X, \quad x\equiv b \;(\operatorname{mod}{W\kappa}) \right\}, \end{equation*}

for some $W,\kappa\in\mathbb{N}$ such that W is divisible by all primes $p\leqslant w$ for some sufficiently large $w\in\mathbb{N}$. If we choose $D,W,\kappa$ appropriately, then we can ensure that the set

\begin{equation*} \left\{\frac{h(x)-h(b)}{\lambda(D)}: b \lt x \leqslant X, \quad x\equiv b \;(\operatorname{mod}{W\kappa}) \right\} \end{equation*}

equidistributes over congruence classes modulo p for any prime $p\leqslant w$. The contribution of the remaining primes is then subsumed by the error term emerging from the transference of solutions from the ‘dense model’ g to ν. The appearance of the additional parameter $\kappa\in\mathbb{N}$ here, resulting in a ‘double W-trick’, was the main innovation of our previous work [Reference Chapman and Chow2]. Its purpose is to ensure that ${\lambda}(D)$ precisely accounts for all common divisors of the values of $h(x) - h(b)$, as x ranges over the arithmetic progression b modulo $W {\kappa}$.

3.1. The W-trick

Consider a set $A \subseteq \mathcal P_X$ with $|A| \geqslant {\delta} |\mathcal P_X|$. Let $C \in \mathbb{N}$ be large with respect to the fixed parameters, and let $w\in\mathbb{N}$ be large in terms of C. Define

\begin{equation*} M = Cd^2 10^{4w}, \qquad W = \left( \prod_{p \leqslant w} p \right) ^{100dw}, \qquad V = \sqrt W \end{equation*}

and

\begin{equation*} D = W^2, \qquad Z = \frac{X - r_D}{D}, \qquad N = h_D(Z) = \frac{h(X)}{{\lambda}(D)}. \end{equation*}

Henceforth, we take $X\in\mathbb{N}$ sufficiently large in terms of $C,w,$ and the fixed parameters. We also assume that $D\mid(X-r_D)$, whence $Z \in \mathbb{N}$.

For $R \in \mathbb{N}$ and $b \in [R]$, we write

\begin{equation*} A_{b,R} := \{x \in A: x \equiv b \;(\operatorname{mod}{R}) \}. \end{equation*}

We denote by $(H,W)_d$ the greatest $m \in \mathbb{N}$ for which $m^d \mid (H,W)$. By [Reference Chapman and Chow2, lemma A.5] and the Siegel–Walfisz theorem, we have

\begin{equation*} \delta|\mathcal P_X| \leqslant |A| \leqslant \sum_{\substack{ b \in [W]: \\ (b,W) = 1 \\ (h'(b),W)_d \leqslant M}} |A_{b,W}| + O\left(10^w W M^{-1/2} \frac{X}{\varphi(W) \log X} \right). \end{equation*}

Since w is large relative to C and d, we have $M \lt 2^{50w}$. Hence, if $(h'(b),W)_d \leqslant M$, then there cannot exist a prime $p\leqslant w$, which divides $(h'(b),W)$ with multiplicity greater than 50dw. It follows that if $(h'(b),W)_d \leqslant M$, then $(h'(b),W)\mid V$. By incorporating the crude estimate

\begin{equation*} \frac{W}{\varphi(W)} = \prod_{p\leqslant w}\left( 1- \frac{1}{p}\right)^{-1} \leqslant 2^w, \end{equation*}

we find that

\begin{equation*} \frac{{\delta} X}{\log X} \ll \sum_{\substack{ b \in [W]: \\ (b,W) = 1 \\ (h'(b),W) \mid V}} |A_{b,W}|. \end{equation*}

Thus, there exists $b_0 \in [W]$ such that

\begin{equation*} |A_{b_0,W}| \gg \frac{{\delta} X}{\varphi(W) \log X}, \qquad (b_0, W) = 1, \qquad (h'(b_0),W) \mid V. \end{equation*}

Define ${\kappa} \in \mathbb{N}$ by

\begin{equation*} W {\kappa} (h'(b_0),W) = {\lambda}(D). \end{equation*}

Note that (2.4) implies that κ is w-smooth, whence $\varphi(W)\kappa = \varphi(W\kappa)$. By pigeonholing, we can then find $b \in [W {\kappa}]$ such that

\begin{equation*} b \equiv b_0 \;(\operatorname{mod}{W}), \qquad |A_{b,W{\kappa}}| \gg \frac{{\delta} X}{\varphi(W) {\kappa} \log X} = \frac{{\delta} X}{\varphi(W{\kappa}) \log X}. \end{equation*}

Since $(h'(b),W) = (h'(b_0), W) \mid V$, we also have

\begin{equation*} (h'(b),W{\kappa}) = (h'(b),W). \end{equation*}

3.2. The weight function

Our next task is to construct an appropriately ‘pseudorandom’ weight function. Let $w,W$, and κ be as defined in the previous subsection. Fix some $b\in[W\kappa]$, which satisfies

(3.1)\begin{equation} (h'(b),W)\mid V=\sqrt{W}. \end{equation}

We then define

(3.2)\begin{equation} \nu: \mathbb{Z} \to [0,\infty), \qquad \nu(n) := \frac{\varphi(W)}{W(h'(b),W)} \sum_{\substack{b \lt p \leqslant X \\ p \equiv b \;(\operatorname{mod}{W {\kappa}}) \\ h(p) - h(b) = n {\lambda}(D)}} h'(p) \log p. \end{equation}

Observe that ν is supported on the set

\begin{equation*} \left \{n\in\mathbb{N}: n=\frac{h(p) - h(b)}{{\lambda}(D)},\; p\in\mathcal P_X,\; p\equiv b \;(\operatorname{mod}{W\kappa}) \right \}\subseteq [N]. \end{equation*}

Recall from the previous subsection that we are considering a fixed set $A\subseteq\mathcal P_X$ with $|A|\geqslant\delta|\mathcal P_X|$, and that we made a judicious choice of $b\in[W\kappa]$ so that $|A_{b,W\kappa}|$ is suitably dense. For this specific choice of b, let

(3.3)\begin{equation} \mathcal A = \left \{ \frac{h(p) - h(b)}{{\lambda}(D)}: p \in A_{b,W{\kappa}} \right \}. \end{equation}

Lemma 3.1. Let $\mathcal A,\nu$ be as defined above. If X is sufficiently large in terms of $h,\delta,w$, then

\begin{equation*} \sum_{n \in \mathcal A} \nu(n) \gg_{\delta} N. \end{equation*}

Proof. Let c be a small, positive constant, and let

\begin{equation*} \Omega_{b,W\kappa,c}(X) = \{p \leqslant c{\delta} X: \: p \equiv b \;(\operatorname{mod}{W {\kappa}}) \}. \end{equation*}

For X sufficiently large, the Siegel–Walfisz theorem implies that $|A_{b,W{\kappa}}| \geqslant |\Omega_{b,W\kappa,c}(X)|$. It follows that there exists an injective map $\psi:\Omega_{b,W\kappa,c}(X)\to A_{b,W{\kappa}}$ such that $p\leqslant\psi(p)$ for all $p\in\Omega_{b,W\kappa,c}(X)$. This implies that

\begin{equation*} \sum_{\substack{p \leqslant c{\delta} X \\ p \equiv b \;(\operatorname{mod}{W {\kappa}})}} p^{d-1} \log p \leqslant \sum_{\substack{p \leqslant c{\delta} X \\ p \equiv b \;(\operatorname{mod}{W {\kappa}})}} \psi(p)^{d-1} \log (\psi(p)) \leqslant \sum_{p \in A_{b,W{\kappa}}} p^{d-1} \log p. \end{equation*}

Invoking the bound $h'(x)\ll x^{d-1}$, we deduce that

\begin{equation*} \sum_{p \in A_{b,W{\kappa}}} h'(p) \log p \gg \sum_{\substack{p \leqslant c{\delta} X \\ p \equiv b \;(\operatorname{mod}{W {\kappa}})}} p^{d-1} \log p. \end{equation*}

By the Siegel–Walfisz theorem again, we thus have

\begin{equation*} \sum_{p \in A_{b,W{\kappa}}} h'(p) \log p \gg_{\delta} \frac{X^d}{\varphi(W {\kappa})}. \end{equation*}

Therefore,

\begin{equation*} \frac{W(h'(b),W)}{\varphi(W)} \sum_{n \in \mathcal A} \nu(n) = O((W{\kappa})^{d-1} \log W) + \sum_{p \in A_{b,W{\kappa}}} h'(p) \log p \gg_\delta \frac{X^d}{\varphi(W {\kappa})}. \end{equation*}

Thus, for our choice of κ and b, the desired bound now follows from the equalities

\begin{equation*} \frac{W(h'(b),W)}{\varphi(W)} = \frac{\lambda(D)}{\kappa \varphi(W)} = \frac{\lambda(D)}{\varphi(W\kappa)}. \end{equation*}

6.1. Restriction I

We begin with the following restriction estimate for ν.

Proposition 6.1. Let $E \gt 2T$, and let $\phi: \mathbb{Z} \to \mathbb{C}$ with $|\phi| \leqslant \nu$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \phi({\alpha})|^E {\,{\rm d}} {\alpha} \ll_{h,E} N^{E-1}. \end{equation*}

This is easily bootstrapped to the following restriction estimate for $\nu + 1_{[N]}$, see the deduction of [Reference Chapman and Chow2, lemma 6.2].

Proposition 6.2. Let $E \gt 2T$, and let $\phi: \mathbb{Z} \to \mathbb{C}$ with $|\phi| \leqslant \nu + 1_{[N]}$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \phi({\alpha})|^E {\,{\rm d}} {\alpha} \ll_{h,E} N^{E-1}. \end{equation*}

To prove proposition 6.1, we proceed in stages. We introduce the auxiliary function

\begin{equation*} \mu: \mathbb{Z} \to [0,\infty), \qquad \mu(n) = \frac{1}{(h'(b),W)} \sum_{\substack{ b \lt x \leqslant X \\ x \equiv b \;(\operatorname{mod}{W {\kappa}}) \\ h(x) - h(b) = n {\lambda}(D)}} h'(x), \end{equation*}

noting that $\nu \leqslant (\log X) \mu$ pointwise. We compute that

\begin{align*} \| \mu \|_1 &= (h'(b),W)^{-1} \sum_{\substack{b \lt x \leqslant X \\ x \equiv b \;(\operatorname{mod}{W {\kappa}})}} h'(x) \\ &\ll (h'(b),W)^{-1} \sum_{y \leqslant X/W {\kappa}} (W {\kappa} y)^{d-1} \\ &\ll \frac{X^d}{W {\kappa} (h'(b),W)} = \frac{X^d}{{\lambda}(D)} \ll N. \end{align*}

We begin with an epsilon-slack restriction estimate for µ.

Lemma 6.3. (Epsilon-slack estimate)

Let $\psi: \mathbb{Z} \to \mathbb{C}$ with $|\psi| \leqslant \mu$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \psi({\alpha})|^{2T} {\,{\rm d}} {\alpha} \ll_{h,\varepsilon} N^{2T-1+\varepsilon}. \end{equation*}

Proof. By orthogonality,

\begin{align*} \int_{\mathbb{T}} |\hat \psi({\alpha})|^{2T} {\,{\rm d}} {\alpha} = \sum_{n_1 + \cdots + n_T = n_{T+1} + \cdots + n_{2T}} \psi(n_1) \cdots \psi(n_T) \overline{ \psi(n_{T+1}) \cdots \psi(n_{2T})}. \end{align*}

As $\| \psi \|_\infty \leqslant \| \mu \|_\infty \ll X^{d-1}$, and since ψ is supported on

\begin{equation*} \left \{ \frac{h(x) - h(b)}{{\lambda}(D)}: x \in [X] \right \}, \end{equation*}

we obtain

\begin{align*} &\int_{\mathbb{T}} |\hat \psi({\alpha})|^{2T} {\,{\rm d}} {\alpha} \\ &\ll (X^{d-1})^{2T} \# \{\mathbf{x} \in [X]^{2T}: h(x_1) + \cdots + h(x_T) = h(x_{T+1}) + \cdots + h(x_{2T}) \}. \end{align*}

Finally, using that $T = T(d)$, we find that

\begin{align*} \int_{\mathbb{T}} |\hat \psi({\alpha})|^{2T} {\,{\rm d}} {\alpha} \ll X^{2T(d-1) + 2T - d + \varepsilon} \ll N^{2T-1+2\varepsilon}. \end{align*}

Our next goal is to largely remove ɛ from the exponent in this restriction estimate for µ, obtaining a log-slack estimate by passing to a slightly higher moment. To accomplish this, we require some bounds on

\begin{equation*} \hat \mu({\theta}) = \frac1{(h'(b),W)} \sum_{\substack{ b \lt x \leqslant X \\ x \equiv b \;(\operatorname{mod}{W {\kappa}})}} h'(x) e\left({\theta} \frac{h(x)-h(b)}{{\lambda}(D)} \right). \end{equation*}

The triangle inequality and partial summation yield

\begin{equation*} \hat \mu({\theta}) \ll \frac{X^{d-1}}{(h'(b),W)} \left( 1 + \max_{X^{1/2} \leqslant P \leqslant X} |g({\theta};P)| \right), \end{equation*}

where

\begin{equation*} g({\theta};P) = \frac1{(h'(b),W)} \sum_{\substack{ b \lt x \leqslant X + b \\ x \equiv b \;(\operatorname{mod}{W {\kappa}})}} e\left({\theta} \frac{h(x)-h(b)}{{\lambda}(D)} \right). \end{equation*}

Writing $x = W{\kappa} y + b$ gives

\begin{equation*} \sum_{\substack{ x \leqslant X \\ x \equiv b \;(\operatorname{mod}{W {\kappa}})}} e\left({\theta} \frac{h(x)-h(b)}{{\lambda}(D)} \right) = \sum_{y \leqslant X/(W{\kappa})} e\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta} f_d(y) \right), \end{equation*}

where c_d is the leading coefficient of h and $f_d(y) \in \mathbb{Z}[y]$ is monic of degree d.

Suppose $X^{1/2} \leqslant P \leqslant X$. The exponential sum

\begin{equation*} g_1({\alpha}; P) := \sum_{y \leqslant P/(W{\kappa})} e({\alpha} f_d(y)) \end{equation*}

can be treated using Roger Baker’s estimates [Reference Baker1]. We apply the formulation [Reference Chow3, lemma 2.3], noting from its proof that the quantity ${\sigma}(d)$ therein can be replaced by $2^{1-d}$. This delivers the following conclusion.

Lemma 6.4. If $|g_1({\alpha}; P)| \gt (P/(W{\kappa}))^{1 - 2^{1-d} + \varepsilon}$, then there exist coprime $r \in \mathbb{N}$ and $b \in \mathbb{Z}$ such that

\begin{equation*} g_1({\alpha}; P) \ll_{d,\varepsilon} r^{\varepsilon-1/d} P(W{\kappa})^{-1} (1 + (P/(W{\kappa}))^d |{\alpha} - b/r|)^{-1/d}. \end{equation*}

Lemma 6.5. Let

\begin{equation*} \mathfrak n = \{{\theta} \in \mathbb{T}: |\hat \mu({\theta})| \leqslant X^{d-2^{-d}} \}. \end{equation*}

If ${\theta} \in \mathbb{T} \setminus \mathfrak n$, then there exist coprime $q \in \mathbb{N}$ and $a \in \mathbb{Z}$ such that

\begin{equation*} \hat \mu({\theta}) \ll_{d,\varepsilon} N (\log X) q^{\varepsilon-1/d} (1 + N|{\theta} - a/q|)^{-1/d}. \end{equation*}

Proof. Let $P \in [X^{1/2}, X]$ maximize

\begin{equation*} \left|g_1\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta}; P \right)\right|. \end{equation*}

By (6.1),

\begin{equation*} \left|g_1\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta}; P \right)\right| \gt (P/(W{\kappa}))^{1 - 2^{1-d} + \varepsilon}. \end{equation*}

By lemma 6.4, there exist coprime $r \in \mathbb{N}$ and $b \in \mathbb{Z}$ such that

\begin{align*} g_1\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta}; P \right) &\ll r^{\varepsilon-1/d} P(W{\kappa})^{-1} \left(1 + (P/(W{\kappa}))^d \left|\frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta} - b/r \right| \right)^{-1/d} \\ &\ll r^{\varepsilon-1/d} X(W{\kappa})^{-1} \left(1 + (X/(W{\kappa}))^d \left|\frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta} - b/r \right| \right)^{-1/d}. \end{align*}

With

\begin{equation*} a = \frac{b} {(b, c_d (W{\kappa})^d/{\lambda}(D))}, \qquad q = \frac{rc_d (W{\kappa})^d/{\lambda}(D)} {(b, c_d (W{\kappa})^d/{\lambda}(D))}, \end{equation*}

we now have

\begin{align*} g_1\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta}; P \right) &\ll r^{\varepsilon-1/d} \frac{X}{W{\kappa}} \left(1 + \frac{X^d}{{\lambda}(D)} |{\theta}-a/q| \right)^{-1/d} \\ &\ll q^{\varepsilon-1/d} \frac{X\log X}{W{\kappa}} \left(1 + \frac{X^d}{{\lambda}(D)} |{\theta}-a/q| \right)^{-1/d}, \end{align*}

since X is large in terms of w. Finally, recall that

\begin{equation*} N \asymp \frac{X^d}{{\lambda}(D)} = \frac{X^d}{W{\kappa}(h'(b),W)} \end{equation*}

and

\begin{equation*} X^{d-2^{-d}} \lt |\hat{\mu}({\theta})| \ll X^{d-1} + \frac{X^{d-1}} {(h'(b),W)} g_1\left( \frac{c_d (W{\kappa})^d}{{\lambda}(D)} {\theta}; P \right). \end{equation*}

By passing to a slightly higher moment, we are now able to obtain a log-slack analogue of lemma 6.3.

Lemma 6.6. (Log-slack estimate)

Let $v \gt 2T$ be real, and let $\psi: \mathbb{Z} \to \mathbb{C}$ with $|\psi| \leqslant \mu$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \psi({\alpha})|^v {\,{\rm d}} {\alpha} \ll_{h,v} N^{v-1} (\log X)^v. \end{equation*}

Proof. Inserting lemmas 6.3 and 6.5 into lemma 2.3 gives

\begin{equation*} \int_{\mathbb{T}} (|\hat \psi({\alpha})| /\log X)^v {\,{\rm d}} {\alpha} \ll N^{v-1}. \end{equation*}

Proof of proposition 6.1

Let $v \in (2T, E)$. Applying lemma 6.6 to $\psi = (\log X)^{-1} \phi$ gives the log-slack restriction estimate

\begin{equation*} \int_{\mathbb{T}} |\hat \phi({\alpha})|^v {\,{\rm d}} {\alpha} \ll_{h,v} N^{v-1} (\log X)^{2v}. \end{equation*}

We can choose σ_d to be large in terms of E and v. Thus, in view of corollaries 5.3 and 5.6, the desired result follows from lemma 2.3.

6.2. Restriction II

Define the auxiliary weight function $\nu_D: \mathbb{Z} \to [0,\infty)$ by

\begin{align*} \nu_D(n) &= \frac{\varphi(D)}{{\lambda}(D)} \sum_{\substack{p \leqslant X \\ p \equiv r_D \;(\operatorname{mod}{D}) \\ h(p) = n {\lambda}(D)}} h'(p) \log p \\ &= \frac{\varphi(D)}{{\lambda}(D)} \sum_{\substack{z \leqslant Z \\ (Dz + r_D)\in\mathcal P_X \\ h_D(z)=n}} h'(Dz + r_D) \log(Dz + r_D) . \end{align*}

Observe that ν_D is supported on $h_D([Z])\subseteq [N]$. By the Siegel–Walfisz theorem, we have

\begin{align*} \| \nu_D \|_1 &= \frac{\varphi(D)}{{\lambda}(D)} \sum_{\substack{p \leqslant X \\ p \equiv r_D \;(\operatorname{mod}{D})}} h'(p) \log p \\ &\leqslant \frac{\varphi(D)}{{\lambda}(D)} \sum_{\substack{p \leqslant X \\ p \equiv r_D \;(\operatorname{mod}{D})}} h'(X) \log X \ll N. \end{align*}

One can be more precise using partial summation, but we do not need to.

In this subsection, we establish the following restriction estimate for ν_D.

Proposition 6.7. Let $E \gt 2T$ be real, and let $\phi: \mathbb{Z} \to \mathbb{C}$ with $|\phi| \leqslant \nu_D$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \phi({\alpha})|^E {\,{\rm d}} {\alpha} \ll_{h,E} N^{E-1}. \end{equation*}

Our approach is similar to that of proposition 6.1, so we will not repeat all of the details. We introduce the auxiliary function

\begin{equation*} \mu_D: \mathbb{Z} \to [0,\infty), \qquad \mu_D(n) = \frac{ND}{X} \sum_{\substack{z \leqslant Z \\ h_D(z) = n}} 1, \end{equation*}

noting that $\nu_D \ll (\log X) \mu_D$ pointwise. As a special case of [Reference Chapman and Chow2, lemma 6.3], we have the following sharp restriction estimate for µ_D.

Lemma 6.8. Let $E \gt 2T$, and let $\psi: \mathbb{Z} \to \mathbb{C}$ with $|\psi| \leqslant \mu_D$. Then,

\begin{equation*} \int_{\mathbb{T}} |\hat \psi({\alpha})|^E {\,{\rm d}} {\alpha} \ll_{h,E} N^{E-1}. \end{equation*}

The upshot is that if $v \gt 2T$ and $|\phi| \leqslant \nu_D$, then

(6.1)\begin{equation} \int_{\mathbb{T}} |\hat \phi({\alpha})|^v {\,{\rm d}} {\alpha} \ll_v N^{v-1} (\log X)^v. \end{equation}

To apply the general epsilon-removal lemma, we require major and minor arc bounds for

\begin{equation*} \hat \nu_D({\alpha}) = \frac{\varphi(D)}{{\lambda}(D)} \sum_{\substack{p \leqslant X \\ p \equiv r_D \;(\operatorname{mod}{D})}} h'(p) \log p \cdot e({\alpha} h(p) / {\lambda}(D)). \end{equation*}

On the minor arcs, we infer the following analogue of corollary 5.3, by essentially the same argument:

(6.2)\begin{equation} \hat \nu_D({\alpha}) \ll X^d (\log X)^{-{\sigma}_d} \qquad ({\alpha} \in \mathfrak m). \end{equation}

On the major arcs, we have (5.2), for some $q,a \in \mathbb{Z}$. Define

\begin{equation*} S_D(q,a) = \sum_{\substack{t \;(\operatorname{mod}{Dq}) \\ (t,q) = 1 \\ t \equiv r_D \;(\operatorname{mod}{D})}} e_q(ah(t)/{\lambda}(D)). \end{equation*}

We infer the following analogue of corollary 5.4 by essentially the same proof.

Lemma 6.9. Let c be a small, positive constant, small in terms of C_d. Suppose $({\alpha}, q, a) \in \mathbb{R} \times \mathbb{N} \times \mathbb{Z}$ with (5.2), and put ${\beta} = {\alpha} - a/q$. Then,

\begin{equation*} \hat \nu_D({\alpha}) = \frac{\varphi(D)} {\varphi(Dq)} S_D(q,a) I({\beta}) + O(Ne^{-c \sqrt{\log X}}). \end{equation*}

It follows from lemma 2.2 that

\begin{equation*} S_D(q,a) \ll_{h,\varepsilon} q^{\varepsilon-1/d} \varphi(q). \end{equation*}

Thus, by incorporating (5.6), we arrive at the following variant of corollary 5.6:

(6.3)\begin{equation} \hat \nu_D({\alpha}) \ll_{h,\varepsilon} q^{\varepsilon - 1/d} \min \{N, \| {\alpha} - a/q \|^{-1} \} + O(Ne^{-c \sqrt{\log X}}). \end{equation}

Equipped with the bounds (6.1), (6.2), and (6.3), proposition 6.7 now follows from the general epsilon-removal lemma (lemma 2.3).

9.1. Exponential sums

To study prime polynomial Bohr sets, we are interested in exponential sums over primes of the form

\begin{equation*} \sum_{\substack{p \leqslant P \\ p \equiv r_D \;(\operatorname{mod}{D})}} e \left( \frac{h(p) \theta}{{\lambda}(D)} \right) \log p, \end{equation*}

where $\theta\in\mathbb{T}$ and h is intersective of the second kind. Lê and Spencer [Reference Lê and Spencer14] analysed properties of sums of this form to obtain estimates for the smallest element of a prime polynomial Bohr set (9.2), showing in particular that these sets are always non-empty. Our goal is to obtain a lower bound for the densities of these Bohr sets which does not depend on the choice of phase α.

Following [Reference Lê and Spencer14], our argument begins with the observation that if the prime polynomial Bohr set (9.2) has few elements, then we can construct a corresponding exponential sum which is particularly large. This is elucidated by the following lemma, which is a consequence of a much more general result of Harman [Reference Harman8].

Lemma 9.3. Let $D,K,P\in\mathbb{N}$. Let h be an integer polynomial of degree $d\in\mathbb{N}$ which is intersective of the second kind. Define r_D and $\lambda(D)$ as in §2.4. Let $\rho\in (0,1)$, $\boldsymbol{\alpha}\in\mathbb{T}^K$, and

\begin{equation*} \mathcal C = \mathcal C_h(\boldsymbol{\alpha},\rho):= \mathcal P\cap\left \{p \leqslant P: p \equiv r_D \;(\operatorname{mod}{D}), \left \| \frac{h(p)}{{\lambda}(D)} \boldsymbol{{\alpha}} \right \| \geqslant \rho \right \}. \end{equation*}

Then, there exists $\mathbf{m}\in\mathbb{Z}^K$ with $0 \lt \lVert \mathbf{m}\rVert_{\infty} \leqslant K\rho^{-1}$ such that

\begin{equation*} (2K + 1)^K\left\lvert \sum_{p\in\mathcal C} e \left(\frac{h(p) \mathbf{m}\cdot\boldsymbol{\alpha}}{{\lambda}(D)} \right) \log p\right\rvert \geqslant \frac{\rho^K}{4K^2 - 1} \sum_{p\in\mathcal C}\log p. \end{equation*}

For the purpose of proving theorem 9.2, we may assume that the Bohr set $\mathcal B_h(\boldsymbol{\alpha},\rho)$ has too few elements, in a manner that will be clarified in due course. Then we can apply lemma 9.3 to find some $L\ll_K \rho^{-K}$ and $\mathbf{m}\in\mathbb{Z}^K$ of bounded size such that $\theta = \mathbf{m}\cdot\boldsymbol{\alpha}$ satisfies

(9.4)\begin{equation} \left| \sum_{\substack{p \leqslant P \\ p \equiv r_D \;(\operatorname{mod}{D})}} e \left( \frac{h(p) \theta}{{\lambda}(D)} \right) \log p \right| \geqslant \frac{P}{L\varphi(D)}. \end{equation}

Our next task is to investigate the consequences of (9.4). As discussed in §4, an exponential sum being large is indicative of the phase θ exhibiting ‘major arc’ behaviour, meaning that θ is well-approximated by a rational number with small denominator. This is made precise in the following lemma.

Lemma 9.4. (Low major arc)

Suppose $L,P\in\mathbb{N}$ and $\theta\in\mathbb{R}$ satisfy (9.4). Assume that P is sufficiently large relative to $D,h$ and L. Then, there exists $q \in \mathbb{N}$ such that

\begin{equation*} q \ll_{h,\varepsilon} L^{d+\varepsilon}, \qquad \| q\theta \| \ll_h qL^d {\lambda}(D) / P^d. \end{equation*}

Proof. By (9.4) and lemma 4.2, we can find $r\in\mathbb{N}$ and $b\in\mathbb{Z}$ such that

\begin{equation*} \max \left \{r, P^d \left| r \frac{\theta} {{\lambda}(D)} - b \right| \right \} \ll (\log P)^{2 C_d}. \end{equation*}

Let $q \in \mathbb{N}$ and $a \in \mathbb{Z}$ with

\begin{equation*} (a,q) = 1, \qquad \frac{{\lambda}(D) b}{r} = \frac a q. \end{equation*}

Then

\begin{equation*} q \leqslant r \ll (\log P)^{2 C_d}, \qquad |q \theta - a| \leqslant |r \theta - {\lambda}(D) b| \ll \frac{{\lambda}(D) (\log P)^{2C_d}} {P^d}. \end{equation*}

Put ${\beta} = \theta - a/q$. Applying lemma 4.3 with $(f,G,b,m,Q) = (h, 1, r_D, D, \lambda(D))$ gives

\begin{equation*} \sum_{\substack{p \leqslant P \\ p \equiv r_D \;(\operatorname{mod}{D})}} e \left( \frac{\theta h(p)}{{\lambda}(D)} \right) \log p = I({\beta}) \frac{S(q,a;D)}{\varphi(Dq)} + O_h(Pe^{-c \sqrt{\log P}}), \end{equation*}

where

\begin{equation*} I({\beta}) = \int_2^P e\left( \frac{{\beta} h(t)}{{\lambda}(D)}\right) {\,{\rm d}} t, \qquad S(q,a;D) = \sum_{\substack{t \;(\operatorname{mod}{Dq}) \\ (t,q) = 1 \\ t \equiv r_D \;(\operatorname{mod}{D})}} e \left( \frac{a h(t)} {q{\lambda}(D)} \right). \end{equation*}

Thus, by (9.4), we have

(9.5)\begin{equation} |S(q,a;D) I({\beta})| \gg \frac{P \varphi(Dq)}{L \varphi(D)}. \end{equation}

In light of the trivial bound $|I(\beta)|\leqslant P$, we now have

\begin{equation*} |S(q,a;D)| \gg \frac{\varphi(Dq)}{L \varphi(D)} \geqslant \frac{\varphi(q)}{L}. \end{equation*}

By [Reference Lucier15, lemma 28], the GCD of the non-constant coefficients of

\begin{equation*} h_D(x) := \frac{h(r_D + Dx)}{{\lambda}(D)} \in \mathbb{Z}[x] \end{equation*}

is $O_h(1)$. Now lemma 2.2 yields

\begin{equation*} S(q,a;D) \ll_{h,\varepsilon} q^{1+\varepsilon-1/d}, \end{equation*}

and we conclude that

\begin{equation*} q \ll_{h,\varepsilon} L^{d+\varepsilon}. \end{equation*}

It remains to bound $\lVert q\theta\rVert$.

Observe that

\begin{align*} \psi: (\mathbb{Z}/Dq\mathbb{Z})^\times &\to (\mathbb{Z}/D\mathbb{Z})^\times \\ [t] &\mapsto [t] \end{align*}

defines a surjective group homomorphism, and that $\psi^{-1}(r_D)$ is a coset of $\ker(\psi) \leqslant (\mathbb{Z}/Dq\mathbb{Z})^\times$. Therefore,

\begin{align*} |S(q,a;D)| &\leqslant |\psi^{-1}(r_D)| = |\ker(\psi)| = \frac{\varphi(Dq)}{\varphi(D)}. \end{align*}

Pairing this with (9.5) yields

\begin{equation*} |I({\beta})| \gg P/L. \end{equation*}

A change of variables gives

\begin{equation*} I({\beta}) = D \int_0^{P/D} e({\beta} h_D(z)) {\,{\rm d}} z + O(D), \end{equation*}

and now [Reference Vaughan22, theorem 7.3] yields

\begin{equation*} I({\beta}) \ll_h (|{\beta}|/{\lambda}(D))^{-1/d} + D. \end{equation*}

Consequently,

\begin{equation*} {\beta} \ll (L/P)^d {\lambda}(D), \end{equation*}

and finally,

\begin{equation*} \| q\theta \| \ll q L^d {\lambda}(D) /P^d. \end{equation*}

9.2. Proof of theorem 9.2

Let $K\in\mathbb{N}$. Writing $\mathcal B=\mathcal B_h(\boldsymbol{\alpha},\rho)$ and $\mathcal C=\mathcal C_h(\boldsymbol{\alpha},\rho)$, we deduce from the Siegel–Walfisz theorem (theorem 2.1) that

\begin{equation*} \sum_{p \in \mathcal B} \log p + \sum_{p \in \mathcal C} \log p = \sum_{\substack{p \leqslant P \\ p \equiv r_D \;(\operatorname{mod}{D})}}\log p \asymp \frac{P}{\varphi(D)}. \end{equation*}

Suppose $\boldsymbol{\alpha}\in\mathbb{T}^K$ is such that

\begin{equation*} \sum_{p \in \mathcal B} \log p \leqslant\frac{c(K)\rho^K P}{\varphi(D)}, \end{equation*}

for some suitably small $c(K) \gt 0$. If no such α were to exist, then theorem 9.2 would hold with $\Delta(h,K,\rho)\gg_K \rho^K$ which, since we demand that $\Delta(h,K,\rho)\leqslant 1$, is stronger than the bound we require. For this choice of α, we infer from lemma 9.3 the existence of some $\mathbf{m}\in\mathbb{Z}^K$ with $0 \lt \lVert \mathbf{m}\rVert_{\infty}\leqslant K\rho^{-1}$ such that

\begin{equation*} \left\lvert \sum_{p\in\mathcal C} e \left(\frac{h(p) \mathbf{m}\cdot\boldsymbol{\alpha}}{{\lambda}(D)} \right) \log p\right\rvert \gg_K \frac{\rho^K P}{\varphi(D)}. \end{equation*}

We also have

\begin{equation*} \left\lvert \sum_{p\in\mathcal B} e \left(\frac{h(p) \mathbf{m}\cdot\boldsymbol{\alpha}}{{\lambda}(D)} \right) \log p\right\rvert \leqslant \sum_{p\in\mathcal B}\log p \ll \frac{c(K)\rho^K P}{\varphi(D)}. \end{equation*}

Choosing c(K) sufficiently small, the triangle inequality furnishes some $L\ll_K \rho^{-K}$ such that (9.4) holds with $\theta=\mathbf{m}\cdot\boldsymbol{\alpha}$. By increasing the value of L if necessary—which only weakens the bound (9.4) that we have obtained—we may assume that $L = C\rho^{-K}$ for some suitably large constant $C=C(h,K) \gt K$ to be specified later. Applying lemma 9.4 supplies us with some $q\in\mathbb{N}$ such that

(9.6)\begin{equation} q \ll_{h,\varepsilon} L^{d+\varepsilon}, \qquad \| q\mathbf{m} \cdot\boldsymbol{\alpha} \| \ll \frac{qL^d {\lambda}(D)} {P^d}. \end{equation}

To complete the proof, beginning from the above deductions, we proceed by induction on K. First, suppose that K = 1 and write $m\in\mathbb{Z}$ in place of $\mathbf{m}\in\mathbb{Z}^K$. As h is intersective of the second kind, any integer $n \equiv r_{qmD} \;(\operatorname{mod}{qmD})$ satisfies

\begin{equation*} h(n) \equiv 0 \;(\operatorname{mod}{{\lambda}(qmD)}), \qquad (n,qmD) = 1. \end{equation*}

Further, by the Siegel–Walfisz theorem and (9.6), we have

\begin{equation*} \sum_{\substack{p \leqslant P/L^2 \\ p \equiv r_{qmD} \;(\operatorname{mod}{qmD})}} \log p \asymp \frac{P/L^2}{\varphi(qmD)} \gg_{h,\varepsilon} \frac{P}{L^{d+3+\varepsilon} \varphi(D)}. \end{equation*}

Since $r_{qmD} \equiv r_D \;(\operatorname{mod}{D})$, every prime p appearing in the sum on the left is congruent to r_D modulo D. Using (2.5), we have

\begin{equation*} qm {\lambda}(D) \mid {\lambda}(qm) {\lambda}(D) = {\lambda}(qmD) \mid h(p) \end{equation*}

for each prime p in the sum, whence

\begin{equation*} \left\| \frac{{\alpha} h(p)}{{\lambda}(D)} \right\| \leqslant \left|\frac{h(p)} {qm {\lambda}(D)}\right| \| q m {\alpha} \| \ll_h \frac{(P/L^2)^d}{q {\lambda}(D)} \frac{q L^d {\lambda}(D)}{P^d} = L^{-d} = (\rho/C)^{d}. \end{equation*}

Taking C sufficiently large, we deduce that $p \in \mathcal B$. We conclude that

\begin{equation*} \sum_{p\in\mathcal B}\log p \geqslant \sum_{\substack{p \leqslant P/L^2 \\ p \equiv r_{qmD} \;(\operatorname{mod}{qmD})}} \log p \gg_{h,\varepsilon} \frac{P}{L^{d+3+\varepsilon} \varphi(D)} \gg_h \frac{\rho^{d+3+\varepsilon}P}{\varphi(D)}, \end{equation*}

as required.

Now suppose $K\geqslant 2$ and assume the induction hypothesis that theorem 9.2 holds with K − 1 in place of K. Write $\mathbf{m}=(\mathbf{m}',m_K)$ and $\boldsymbol{\alpha}=(\boldsymbol{\alpha}',\alpha_K)$. The induction hypothesis, applied to

\begin{equation*} \frac{{\lambda}(q m_K)}{m_K} \boldsymbol{{\alpha}}', \end{equation*}

informs us that the set

\begin{equation*} \mathcal A = \left \{ p \leqslant P/L^2: p \equiv r_{qm_K D} \;(\operatorname{mod}{qm_K D}), \; \left \| \frac{h(p)}{m_K {\lambda}(D)} \boldsymbol{{\alpha}}' \right \| \lt \frac{\rho^2}{2K^2} \right \} \end{equation*}

satisfies

\begin{align*} \sum_{p\in\mathcal A}\log p &\geqslant \frac{P}{L^2\varphi(qm_K D)} \Delta \left(h,K-1; \frac{\rho^2}{2K^2}\right) \\ &\gg_{h,K,\varepsilon} \frac{P}{\varphi(D)} \rho^{3 + K(d+\varepsilon)} \Delta \left(h,K-1; \frac{\rho^2}{2K^2}\right). \end{align*}

Let $a\in\mathbb{Z}$ be such that

\begin{align*} \lVert q\mathbf{m}\cdot\boldsymbol{\alpha}\rVert = |q\mathbf{m}\cdot \boldsymbol{\alpha} - a|. \end{align*}

For each $p\in\mathcal A$, we have $h(p) \ll_h (P/L^2)^d$ so, by (9.6),

\begin{equation*} \left| {\alpha}_K \frac{h(p)}{{\lambda}(D)} - \frac{a/q - \mathbf{m}' \cdot \boldsymbol{{\alpha}}'}{m_K} \frac{h(p)}{{\lambda}(D)} \right| = \left\lvert \frac{h(p)}{m_K\lambda(D)}\right\rvert\cdot \lvert\mathbf{m}\cdot\boldsymbol{\alpha} - a/q\rvert \ll_h L^{-d}. \end{equation*}

As in the previous case, we have $qm_K\lambda(D) \mid h(p)$, whence

\begin{equation*} \left\lVert \frac{a/q - \mathbf{m}' \cdot \boldsymbol{{\alpha}}'}{m_K} \frac{h(p)}{{\lambda}(D)}\right\rVert = \left\lVert \frac{h(p)\mathbf{m}' \cdot \boldsymbol{{\alpha}}'}{m_K{\lambda}(D)}\right\rVert \lt \frac{\rho}{2}. \end{equation*}

Thus, by the triangle inequality,

\begin{equation*} \left \| {\alpha}_K \frac{h(p)}{{\lambda}(D)} \right \| \lt \frac{\rho}2 + O_h(L^{-d}) = \frac{\rho}2 + O_h((C\rho^{-K})^{-d}). \end{equation*}

By taking C sufficiently large, we find that $p\in\mathcal B$. Therefore

\begin{equation*} \sum_{p\in\mathcal B}\log p \geqslant \sum_{p\in\mathcal A}\log p \gg_{h,K,\varepsilon} \frac{P}{\varphi(D)} \rho^{3 + K(d+\varepsilon)} \Delta \left(h,K-1; \frac{\rho^2}{2K^2}\right), \end{equation*}

as required.