Proof. For $u(x) = \sum _{k \in \mathcal {S}^{d-1}_\lambda } \widehat {u}_k e^{2\pi i k\cdot x}$ in $E_\lambda $ , write

$$ \begin{align*} \int_{\mathbb{T}^2} |u|^2 \mathop{}\!\mathrm{d} \mu = \sum_{k,\ell \in \mathcal{S}^1_\lambda} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}. \end{align*} $$

The trace inequality: If $\mathop {}\!\mathrm {d} \mu = f \mathop {}\!\mathrm {d} x$ with f in $L^2$ , then $(\widehat {\mu }_k)_{k \in \mathbb {Z}^2} \in \ell ^2$ by Parseval’s theorem. Furthermore, any $\xi \in \mathbb {Z}^2$ can be represented as $k-\ell $ with $k,\ell \in \mathcal {S}^1_\lambda $ in at most two different ways. With the Cauchy–Schwarz inequality, this gives

$$ \begin{align*} \biggl| \sum_{k,\ell \in \mathcal{S}^1_\lambda} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| \leq \biggl( \sum_{k,\ell \in \mathcal{S}^1_\lambda} |\widehat{\mu}_{k-\ell}|^2 \biggr)^{1/2} \biggl( \sum_{k,\ell \in \mathcal{S}^1_\lambda} |\widehat{u}_k|^2 |\widehat{u}_\ell|^2 \biggr)^{1/2} \leq \sqrt{2} \| f \|_{L^2} \| u \|_{L^2}^2. \end{align*} $$

If $|\widehat {\mu }_k| \lesssim |k|^{-\epsilon }$ , we use Jarnik’s lemma (Lemma 3.6 or, more precisely, (3.1)) which gives a splitting of $\mathcal {S}^1_\lambda $ into sets $(\Omega _\alpha )$ such that $\# \Omega _\alpha \leq 2$ and $\mathrm {dist}(\Omega _\alpha ,\Omega _\beta ) \gtrsim \lambda ^{1/3}$ for $\alpha \neq \beta $ . Write

(4.2) $$ \begin{align} \sum_{k,\ell \in \mathcal{S}^1_\lambda} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} = \sum_\alpha \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} + \sum_{\alpha \neq \beta} \sum_{k \in \Omega_\alpha} \sum_{\ell \in \Omega_\beta} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}. \end{align} $$

By the Cauchy–Schwarz inequality and the bound $\sharp \mathcal {S}^1_\lambda = N_1(\lambda ) \lesssim _\epsilon \lambda ^{\epsilon /4}$ for any $\epsilon>0$ , we deduce

$$ \begin{align*} \biggl| \sum_{k,\ell \in \mathcal{S}^1_\lambda} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| & \leq \biggl|\sum_\alpha \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| + \biggl| \sum_{\alpha \neq \beta} \sum_{k \in \Omega_\alpha} \sum_{\ell \in \Omega_\beta} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| \\ & \lesssim_\epsilon \sum_\alpha \sum_{k,\ell \in \Omega_\alpha} | \widehat{u}_k \widehat{u}_\ell | + \lambda^{-\epsilon/3} \sum_{k,\ell \in \mathcal{S}^1_\lambda } |\widehat{u}_k \widehat{u}_\ell| \\ & \lesssim \sum_\alpha \sup_\alpha \#\Omega_\alpha \sum_{k \in \Omega_\alpha} |\widehat{u}_k|^2 + \lambda^{-\epsilon/3} N_1(\lambda) \sum_{k \in \mathcal{S}^1_\lambda} |\widehat{u}_k|^2 \lesssim \| u \|_{L^2}^2. \end{align*} $$

The observability inequality: Similarly, we reuse (4.2) and estimate

$$ \begin{align*} \biggl| \sum_{\alpha \neq \beta} \sum_{k \in \Omega_\alpha} \sum_{\ell \in \Omega_\beta} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| \lesssim \begin{cases} \sum_{|k| \gtrsim \lambda^{-1/3}} |\widehat{\mu}_k|^2, & \mu \in L^2; \\ C_\epsilon \lambda^{\epsilon/4} \lambda^{ - \epsilon/3}, & |\widehat{\mu}_k| \lesssim |k|^{-\epsilon}. \end{cases} \end{align*} $$

The important point is that the right-hand side goes to zero as $\lambda \to \infty $ . There remains the first sum in the above equation. We analyze the following two cases separately:

• • If $\Omega _\alpha = \{p\}$ is a singleton, then clearly

  $$ \begin{align*} \sum_\alpha \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} = |\widehat{u}_p|^2. \end{align*} $$

• • If $\Omega _\alpha = \{p,q\}$ is a doubleton, then

  $$ \begin{align*} \sum_\alpha \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} & = |\widehat{u}_p|^2 + |\widehat{u}_q|^2 + \widehat{\mu}_{p-q} (\widehat{u}_p \overline{\widehat{u}_q} + \widehat{u}_q \overline{\widehat{u}_p}) \\ & \ge (1-|\widehat{\mu}_{p-q}|) (|\widehat{u}_p|^2 + |\widehat{u}_q|^2). \end{align*} $$

  This estimate is uniform among all such $\Omega _\alpha $ since by Lemma 3.7, we have

  $$ \begin{align*} 1-|\widehat{\mu}_{p-q}| \ge 1 - \sup_{k \in \mathbb{Z}^2 \setminus 0} |\widehat{\mu}_k|> 0. \end{align*} $$

This proves the semiclassical observability for both cases $(i)$ and $(ii)$ . In the case $(ii)$ , there remains to prove the observability for eigenfunctions with bounded eigenvalues. This is immediate, since trigonometric polynomials cannot vanish on sets of positive Lebesgue measure.

Proof. Choosing $c_0> 0$ as above. By Lemma 3.2, when $r>0$ is sufficiently small, there exists $\lambda \in [ c_0 r^{-1}, 2 c_0 r^{-1} ] \cap \sqrt {\mathbb {N}}$ such that $N_d(\lambda ) \gtrsim _d \lambda ^{d-2}$ . For such $\lambda $ and $x_0 \in \mathbb {T}^d$ , define $\varphi _{\lambda ,x_0} \in E_\lambda $ as in (5.1). By (5.2) and the trace inequality for $\mu $ , there holds

(5.3) $$ \begin{align} \begin{aligned} \mu \bigl(B_{r} (x_0)\bigr) = \int_{B_r (x_0)} \mathop{}\!\mathrm{d} \mu & \lesssim \frac{1}{N_d(\lambda)^2} \int_{\mathbb{T}^d} |\varphi_{\lambda,x_0} |^2 \mathop{}\!\mathrm{d}\mu \\ & \lesssim_\mu \frac{1}{N_d(\lambda)^2} \int_{\mathbb{T}^d} |\varphi_{\lambda,x_0}|^2 \mathop{}\!\mathrm{d} x = \frac{1}{N_d(\lambda)} \lesssim \frac{1}{\lambda^{d-2}} \lesssim_d r^{d-2}. \end{aligned} \end{align} $$

Applying Frostman’s Lemma we conclude the proof.

Proof. We follow a line of argument already appearing in [Reference Germain, Moyano and Zhu23]. Let $P_n$ be the product of all prime numbers which are $\equiv 1 \pmod {4}$ and $\le n$ . By [Reference McCurley34] and [Reference McCurley35], we have $\ln P_n \sim n$ . For $r> 0$ , choose n such that $\sqrt {P_{n-1}} \le 1/r\le \sqrt {P_n}$ , then $n \sim \ln (1/r)$ . Then there holds

(5.4) $$ \begin{align} \sqrt{P_n} \le \frac{1}{r} \frac{\sqrt{P_n}}{\sqrt{P_{n-1}}} \le \frac{\sqrt{n}}{r} \lesssim \frac{\sqrt{\ln(1/r)}}{r}. \end{align} $$

Next, let $\pi _n$ be the number of prime numbers $\le n$ which are $\equiv 1 \pmod {4}$ and recall that, by the prime number theorem for arithmetic progressions, $\pi _n \sim n/\ln n$ .

Let $r>0$ be sufficiently small and let $\lambda = \sqrt {P_n}$ . By (5.4), we have $1/r \le \lambda \lesssim \sqrt {\ln (1/r)}/r$ . Let $\rho = r/\sqrt {\ln (1/r)}$ . We claim that, when $r \le e^{-1}$ (which is always the case since r is sufficiently small), there holds $\rho \le r \le \rho \sqrt {\ln (1/\rho )}$ . Indeed, by definition $\rho \le r/\sqrt {\ln e} = r$ and $r = \rho \sqrt {\ln (1/r)} \le \rho \sqrt {\ln (1/\rho )}$ . Combining these estimates gives

$$ \begin{align*} \frac{1}{\rho \sqrt{\ln(1/\rho)}} \le \lambda \lesssim \frac{1}{\rho}. \end{align*} $$

Therefore, by Jacobi (see [Reference Grosswald25]) and the definition of the primorial $P_n$ , letting $d_j(n)$ be the number factors of n that are $\equiv j \pmod {4}$ , then for some $C,C',C">0$ ,

(5.5) $$ \begin{align} \begin{aligned} N_2(\lambda) & = 4 \bigl(d_1(P_n) - d_3(P_n)\bigr) = 4 d_1(P_n) \\ & = 4 \times 2^{\pi_n} \ge 4 \times 2^{C n/\ln n} \gtrsim \lambda^{C'/\ln\ln\lambda} \ge \rho^{-C"/\ln\ln(1/\rho)}. \end{aligned} \end{align} $$

For $x_0 \in \mathbb {T}^2$ , define $\varphi _{\lambda ,x_0}$ as in (5.1). Fix a sufficiently small $\delta>0$ (independent of r or $\lambda $ ) such that, by (5.2), if $x \in B_{\delta \rho }(x_0)$ , then $|\phi _{\lambda ,x_0}(x)| \gtrsim N_2(\lambda )$ . Arguing as in (5.3) gives

$$ \begin{align*} \mu \bigl(B_{\delta\rho} (x_0)\bigr) = \frac{1}{N_2(\lambda)} \lesssim \rho^{-C"/\ln\ln(1/\rho)}.\\[-45pt] \end{align*} $$

Proof. Let $\lambda _n$ satisfy (5.7) and define $\varphi _n = \varphi _{\lambda _n,x_{\lambda _n}}$ as in (5.1). By (5.2), there exists $\delta \in (0,1)$ such that that $|\varphi _n(x)| \gtrsim N_d(\lambda _n)$ for all $x\in B_{\delta \lambda _n^{-1}}(x_{\lambda _n})$ . Therefore, as $n \to \infty $ ,

$$ \begin{align*} \frac{1}{N_d(\lambda_n)} \int_{\mathbb{T}^d} |\varphi_n|^2 \mathop{}\!\mathrm{d} \mu \gtrsim N_d(\lambda_n) \mu(B_{\delta \lambda_n^{-1}}(x_{\lambda_n} )) \gtrsim_\delta \ln \ln \lambda_n \to \infty. \end{align*} $$

Proof. For $n \ge 0$ , let $\lambda _n = \sqrt {P_n}$ where the primorial $P_n$ is defined in the proof of Proposition 5.2. By (5.5), we have $\lim _{n \to \infty } N_2(\lambda _n) = \infty $ . For $x = (x_1,\ldots ,x_d) \in \mathbb {T}^d$ , put

$$ \begin{align*} g_n(x) = \sum_{(k_1,k_2) \in \mathcal{S}^1_\lambda} e^{2\pi i (k_1 x_1 + k_2 x_2)}. \end{align*} $$

Then $\| g_n \|_{L^2} = \sqrt {N_2(\lambda _n)}$ and $g_n(x) = N_2(\lambda _n)$ for $x \in S$ . We conclude with

$$ \begin{align*} \lim_{n \to \infty} \frac{\int_{\mathbb{T}^{d}} |g_n(x)|^2 \mathop{}\!\mathrm{d} \mu }{\int_{\mathbb{T}^d} |g_n(x)|^2 \mathop{}\!\mathrm{d} x} = \lim_{n \to \infty} N_2(\lambda_n) = \infty.\end{align*} $$

Proof of Proposition 6.1

Since S is compact, there exists $\epsilon _0> 0$ such that for any $x_0 \in S$ , the submanifold S is parametrized as $S_{x_0} = \{(x',\varphi (x')) : x' \in \epsilon _0 B\}$ near $x_0$ , where $B \subset \mathbb {R}^\beta $ is the unit ball centered at the origin, and $\varphi : \mathbb {R}^\beta \to \mathbb {R}^{d-\beta }$ a smooth and $x_0$ -dependent function.

For $k \in \mathcal {S}^{d-1}_\lambda $ , the trace of the eigenfunction $e^{2\pi i k\cdot x}$ on $S_{x_0}$ is $e^{2\pi i k \cdot (x',\varphi (x'))}$ . Next, let $\chi \in C_c^\infty (\mathbb {R}^\beta )$ be such that it equals to one in the unit ball B and is supported in $2B$ . When $0 < \epsilon < \frac {1}{2} \epsilon _0$ , for $a_k \in \mathbb {C}$ where $k \in \mathbb {S}^{d-1}_\lambda $ , one defines $g_\lambda \in C_c^\infty (\mathbb {R}^\beta )$ as

$$ \begin{align*} g_\lambda(x') = \chi \Bigl(\frac {x'} {\epsilon}\Bigr) \sum_{k \in \mathcal{S}^{d-1}_{\lambda}} a_k e^{2\pi i k \cdot (x',\varphi(x'))}. \end{align*} $$

Since $\operatorname {\mathrm {supp}} g_\lambda \subset 2\epsilon B$ , we may naturally regard it as a $4\epsilon \mathbb {Z}^\beta $ -periodic function, so that it lives henceforth in $\mathbb {T}^\beta _{4\epsilon } = \mathbb {R}^\beta / 4\epsilon \mathbb {Z}^\beta $ . Expanding it in Fourier series, one writes

$$ \begin{align*} g_\lambda(x') = (4\epsilon)^{-\beta} \sum_{\ell \in \mathbb{Z}^\beta} e^{2\pi i \frac{\ell}{4\epsilon} \cdot x'} \widehat{g}_\lambda(\ell), \quad \widehat{g}_\lambda(\ell) = \sum_{k \in \mathcal{S}^{d-1}_\lambda} a_k A^\epsilon_\lambda(k,\ell), \end{align*} $$

where the linear coefficients are

$$ \begin{align*} A^\epsilon_\lambda(k,\ell) = \int_{\mathbb{T}^\beta_{4\epsilon}} \chi \Bigl(\frac {x'} {\epsilon}\Bigr) e^{-2\pi i \frac{\ell}{4\epsilon} \cdot x'} e^{2\pi i k \cdot (x',\varphi(x'))} \mathop{}\!\mathrm{d} x'. \end{align*} $$

We will fix parameters $0 < \epsilon < \eta $ depending solely on d and $\beta $ . Then we will find an increasing sequence of $\lambda _n$ and a sequence of nontrivial constants $(a^{(n)}_k:k \in \mathcal {S}^{d-1}_{\lambda _n})$ such that the Fourier coefficients $\widehat {g}_{\lambda _n}(\ell )$ vanish when $|\ell | \le \eta \lambda _n$ . This leads to solving the linear systems:

(6.3) $$ \begin{align} \widehat{g}_{\lambda_n}(\ell) = \sum_{k \in \mathcal{S}^{d-1}_{\lambda_n}} a_k^{(n)} A^\epsilon_{\lambda_n}(k,\ell) = 0, \quad \ell \in \mathbb{Z}^\beta, \ |\ell| \le \eta \lambda_n. \end{align} $$

The number of equations is $\lesssim _\beta \eta ^\beta \lambda ^\beta $ and the number unknowns (which are $(a^{(n)}_k:k \in \mathcal {S}^{d-1}_{\lambda _n})$ ) is $N_d(\lambda _n)$ . By Lemma 3.2, we may choose $\lambda _n$ such that $N_d(\lambda _n) \gtrsim _d \lambda _n^{d-2}$ . Since $\beta \le d-2$ , if $\eta $ is small and n is large, the system (6.3) admits a nontrivial solution $(a_k^{(n)}:k \in \mathcal {S}^{d-1}_{\lambda _n})$ , which we may further assume to be normalized in $\ell ^2(\mathcal {S}^{d-1}_{\lambda _n})$ .

Fix these solutions. By Lemma 6.4 below, since $\epsilon < \eta $ , if $|\ell |> \eta \lambda _n$ , then for all $N \ge 0$ ,

$$ \begin{align*} |\widehat{A}^\epsilon_{\lambda_n}(k,\ell)| \lesssim_{d,\beta,\varphi,\epsilon,\eta,N} \ell^{-3N}. \end{align*} $$

For $N \geq d$ , by the Cauchy–Schwartz inequality, this leads to

$$ \begin{align*} |\widehat{g}_{\lambda_n}(\ell)| \le \Bigl( \sum_{k \in \mathcal{S}^{d-1}_{\lambda_n}} |A_{\lambda_n}^\epsilon(k,\ell)|^2\Bigr)^{1/2} \lesssim_{d,\beta,\varphi,\epsilon,\eta,N} |\ell|^{-3N} N_d(\lambda_n) \lesssim_{d,\eta} |\ell|^{-2N}. \end{align*} $$

Recalling that $\widehat {g}_{\lambda _n} (\ell ) =0$ when $|\ell | \leq \eta \lambda _n $ , by Parseval’s identity we deduce

$$ \begin{align*} \| g_{\lambda_n} \|_{L^2 (\mathbb{T}^\beta_{4\epsilon})}^2 \sim_{\beta,\epsilon} \sum_{|\ell| \geq \eta \lambda_n } |\widehat{g}_{\lambda_n} (\ell)|^2 \lesssim_{d,\beta,\varphi,\epsilon,\eta,N} \lambda_n^{- 4N + \beta} \le \lambda_n^{-N}. \end{align*} $$

Finally, letting $u_n(x) = \sum _{k \in \mathcal {S}^{d-1}_{\lambda _n}} a_k^{(n)} e^{2\pi i k\cdot x} \in E_{\lambda _n}$ , then

$$ \begin{align*} \| u_n \|_{L^2(\mu_{x_0,\epsilon})}^2 \lesssim_{\beta,\epsilon} \| g_{\lambda_n} \|_{L^2 (\mathbb{T}^\beta_{4\epsilon})}^2 \lesssim_{d,\beta,\varphi,\epsilon,\eta,N} \lambda_n^{-N}.\\[-41pt] \end{align*} $$

Proof. Write $k = (k',k") \in \mathbb {Z}^\beta \times \mathbb {Z}^{d-\beta }$ and let $\psi ^\epsilon _{k,\ell }(x') = (k' -\frac {\ell }{4\epsilon }) \cdot x' + k" \cdot \varphi (x')$ . Then

$$ \begin{align*} A_\lambda^\epsilon(k,\ell) = \int_{\mathbb{T}^\beta_{4\epsilon}} \chi\Bigl(\frac{x'}{\epsilon}\Bigr) e^{2\pi i \psi^\epsilon_{k,\ell}(x')} \mathop{}\!\mathrm{d} x'. \end{align*} $$

Since $|k'| \le |k| = \lambda < \frac {|\ell |}{\eta } $ , to conclude, we integrate by part and use the estimate:

$$ \begin{align*} |\nabla_{x'} \psi^\epsilon_{k,\ell} (x')| = \Bigl|\Bigl(k' - \frac \ell \epsilon, k" \cdot \nabla_{x'} \varphi\Bigr) \Bigr| \geq \frac{|\ell|} \epsilon - |k'| \geq |\ell| \Bigl(\frac{1}{\epsilon} - \frac{1}{\eta}\Bigr).\\[-42pt] \end{align*} $$

Proof of Proposition 6.2

By the compactness of $\mathcal {S}$ , there exists $\epsilon _0> 0$ and a finite open covering $S = \bigcup _{j=1}^K S_j$ where, for each j, there exists a smooth function $\varphi _j : \mathbb {R}^\beta \to \mathbb {R}^{d-\beta }$ such that ${S_j = \{(x',\varphi _j(x') : x' \in \epsilon _0 B) \}}$ under a certain system of coordinates.

We proceed as in the proof of Proposition 6.1 and let

$$ \begin{align*} g^j_\lambda(x') = \chi \Bigl(\frac {x'} {\epsilon}\Bigr) \sum_{k \in \mathcal{S}^{d-1}_{\lambda}} a_k e^{2\pi i k \cdot (x',\varphi_j(x'))} = (4\epsilon)^{-\beta} \sum_{\ell \in \mathbb{Z}^\beta} e^{2\pi i \frac{\ell}{4\epsilon} \cdot x'} \widehat{g}^j_\lambda(\ell). \end{align*} $$

Fix an increasing sequence $(\lambda _n)_{n \ge 0}$ such that

• • If $d=3,4$ , then $N_d(\lambda _n) \gtrsim \lambda _n \ln \ln \lambda _n$ ;

• • If $d\ge 5$ , then $N_d(\lambda _n) \gtrsim _d \lambda _n^{d-2}$ .

Setting $\eta = 1$ , we solve the following linear systems:

$$ \begin{align*} \widehat{g}^j_{\lambda_n}(\ell) = 0, \quad \ell \in \mathbb{Z}^\beta,\ |\ell| \le \lambda_n, \ 1 \le j \le K. \end{align*} $$

Now the number of equation is $\lesssim _\beta K \lambda _n^\beta $ . It is smaller than $N_d(\lambda _n)$ , the number of the unknowns (which are $(a_k)$ ), under the assumption of the proposition (either $d=3,4$ and $\beta \le d-2$ or $d \ge 5$ and $\beta \le d-3$ ). Hence, for large n, we find an $\ell ^2$ normalized solution $(a^n_k)_{k \in \mathcal {S}^{d-1}_{\lambda _n}}$ to this system.

To conclude, one defines the eigenfunction $u_n \in E_{\lambda _n}$ as previously, apply the previously obtained estimates of $u_n$ on each coordinate patch, and combines these estimate:

$$ \begin{align*} \|u_n\|_{L^2(\mathop{}\!\mathrm{d}\mu)}^2 \le \sum_{1 \le j \le K} \| \boldsymbol{1}_{S_j} u_n \|_{L^2(\mathop{}\!\mathrm{d}\mu)}^2 \lesssim_{\beta,\epsilon} \sum_{1 \le j \le K} \| g^j_{\lambda_n} \|_{L^2 (\mathbb{T}^\beta_{4\epsilon})}^2 \lesssim_{d,\beta,\varphi,\epsilon,\eta,N} K \lambda_n^{-N}. \end{align*} $$

Proof. Up to a translation, we may assume that $x_0 = 0$ . Let $\chi \in C^\infty _c(\mathbb {R}^d)$ be supported in $\frac {1}{2} B = \{x \in \mathbb {R}^d : |x| \le \frac {1}{2}\}$ . For $u \in C^\infty (\mathbb {T}^d)$ , let

$$ \begin{align*} g(y) = \chi (y) u \left(\frac{y}{\lambda} \right) = \chi(y) \sum_{k \in \mathbb{Z}^d} \widehat{u}_k e^{2 \pi i k \cdot y/\lambda}. \end{align*} $$

Since $\operatorname {\mathrm {supp}} g \subset \frac {1}{2} B$ , we may extend it periodically to $\mathbb {T}^d$ and still denote this extension by g. Let $(\widehat {g}_\ell )_{\ell \in \mathbb {Z}^d}$ be Fourier coefficients of g and let $\widehat {\chi }$ be the Fourier transform of $\chi $ in $\mathbb {R}^d$ . Then, define

$$ \begin{align*} \alpha_{\lambda,\ell,x_0}(u) = \widehat{g}_\ell = \sum_{k \in \mathbb{Z}^d} \widehat{u}_k \int_{\mathbb{T}^d} \chi(y) e^{2\pi i (k/\lambda - \ell)\cdot y} \mathop{}\!\mathrm{d} y = \sum_{k \in \mathbb{Z}^d} \widehat{u}_k \widehat{\chi}\Bigl(\ell-\frac{k}{\lambda}\Bigr). \end{align*} $$

Now assume that $\widehat {u}_k = 0$ when $|k|> \lambda $ . By the Cauchy–Schwartz inequality and the rapid decay of $\widehat {\chi }$ , one deduces that, for all $N \ge 0$ ,

$$ \begin{align*} |\widehat{g}_\ell|^2 \lesssim_{\chi,N} \|u\|_{L^2}^2 \sum_{|k|\le \lambda} \Bigl(1 + \Bigl|\ell - \frac{k}{\lambda}\Bigr|^2\Bigr)^{-N} \le \lambda^d (1+|\ell|)^{-2N} \|u\|_{L^2}^2. \end{align*} $$

This implies that, for all $R \ge 1$ , we have

$$ \begin{align*} \sup_{4\lambda |x| \le 1} \biggl|u(x)-\sum_{|\ell| \le R} \widehat{g}_\ell e^{2\pi i \ell \cdot \lambda x}\biggr| \le \biggl\|g(y) - \sum_{|\ell| \le R} \widehat{g}_\ell e^{2\pi i \ell \cdot y}\biggr\|_{L^\infty} \le \sum_{|\ell|> R} |\widehat{g}_\ell| \lesssim_{\chi,d,N} \lambda^{d/2} R^{-N}. \end{align*} $$

Proof of Proposition 6.6

Let $\phi $ be a Schwartz function in $\mathbb {R}^d$ such that $\widehat {\phi } \in C_c^\infty (\mathbb {R}^d)$ and satisfies $\widehat {\phi }(\xi )=1$ for $|\xi | \le 2$ . For $\lambda \ge 1$ , let $\Phi \in C^\infty (\mathbb {T}^d)$ be such that $\widehat {\Phi }_k = \widehat {\phi }(k/\lambda )$ for $k \in \mathbb {Z}^d$ . Particularly $\widehat {\Phi }_k = 1$ when $|k| \le 2\lambda $ . Consequently, if $u \in E_\lambda $ , then

$$ \begin{align*} \int_{\mathbb{T}^d} |u|^2 \mathop{}\!\mathrm{d}\mu = \int_{\mathbb{T}^d} \Phi * (|u|^2) \mathop{}\!\mathrm{d}\mu = \int_{\mathbb{T}^d} |u|^2 \mathop{}\!\mathrm{d}(\Phi * \mu). \end{align*} $$

By the Poisson summation formula and Fourier rescaling properties, one has

$$ \begin{align*} \Phi(x) = \sum_{k \in \mathbb{Z}^d} \widehat{\phi}\Bigl( \frac{k}{\lambda} \Bigr) e^{2\pi i k\cdot x} = \lambda^d \sum_{q \in \mathbb{Z}^d} \phi(\lambda (x-q)). \end{align*} $$

From this, one immediately sees that $\Phi $ is concentrated near $\mathbb {Z}^d$ (i.e. $0 \in \mathbb {T}^d$ ). Quantitatively, letting $B_{\lambda ^{-1+\epsilon } }(0) = \{x \in \mathbb {T}^d : \mathrm {dist}(x,0) \le \lambda ^{-1+\epsilon }\}$ , then the following estimates hold:

• • $|\Phi (x)| \lesssim _{\phi ,d} \lambda ^d $ for $x \in B_{\lambda ^{-1+\epsilon } }(0)$ ;

• • $|\Phi (x)| \lesssim _{\phi ,d,N, \epsilon } \lambda ^{-N} $ for $N \ge 0$ and $x \notin B_{\lambda ^{-1+\epsilon } }(0)$ .

Therefore, to conclude the proof it is enough to show that there exist $\epsilon>0$ and a sequence of $L^2$ -normalized eigenfunctions $u_n \in E_{\lambda _n}$ with $\lambda _n \to \infty $ such that

$$ \begin{align*} \lim_{n \to \infty} \int_{\mathbb{T}^d} |u_n|^2 \mathop{}\!\mathrm{d} \mu_n^\epsilon = 0, \quad \text{ where } \mu_n^\epsilon = \lambda_n^d \cdot \mathbf{1}_{B_{\lambda_n^{-1+\epsilon} }(0)} * \mu. \end{align*} $$

Let $\beta = \overline {\dim }_{\mathrm {M}} (\operatorname {\mathrm {supp}} \mu )$ and choose $\epsilon> 0$ such that $\beta + 2d\epsilon < d-2$ . Then, from the Minkowski dimension definition, we note that there exists $K \lesssim _{d,\mu } \lambda ^{\beta + d\epsilon }$ and $(x_j)_{j=1}^K$ in $\mathbb {T}^d$ such that

$$ \begin{align*} \operatorname{\mathrm{supp}} \mu_n^\epsilon \subset B_{\lambda_n^{-1+\epsilon} }(0) + \operatorname{\mathrm{supp}} \mu \subset \bigcup_{1 \le j \le K} B_{\lambda_n^{-1}/6}(x_j). \end{align*} $$

Let $\alpha _{\lambda ,\ell ,x_j}$ be defined as in Lemma 6.7. For any $n \in \mathbb {N}$ we can find an $L^2$ -normalized eigenfunction $u^{(n)} \in E_{\lambda _n}$ satisfying the following conditions:

(6.4) $$ \begin{align} \alpha_{\lambda_n,\ell,x_j}(u^{(n)}) = 0, \quad 1 \le j \le K, \ |\ell| \le \lambda^\epsilon. \end{align} $$

Interpreting (6.4) as a linear system in the unknowns $(\widehat {u}^{(n)}_k)_{k \in \mathcal {S}^{d-1}_{\lambda _n}}$ , the number of unknowns is $N_d(\lambda _n)$ , which satisfies $N_d(\lambda _n) \gtrsim \lambda _n^{d-2}$ for a suitably chosen sequence $(\lambda _n)$ (see Lemma 3.2). In contrast, the number of equations satisfies the bound $\lesssim _d K \lambda _n^{d\epsilon } \lesssim _{d,\mu } \lambda _n^{\beta + 2d\epsilon }$ . Since $\beta + 2d\epsilon < d-2$ , for this sequence and sufficiently large n, a nontrivial solution always exists.

For such $u^{(n)}$ , by Lemma 6.7, for all $j \in \{1,\ldots ,K\}$ and $N \ge 0$ , there holds $|u^{(n)}(x)| \lesssim _{d,N} \lambda _n^{d/2-\epsilon N}$ for $x \in B_{\lambda _n^{-1}/6}(x_j)$ . Therefore, taking N large, one obtains as $n \to \infty $ :

$$ \begin{align*} \int_{\mathbb{T}^d} |u_n|^2 \mathop{}\!\mathrm{d} \mu_n^\epsilon & \leq \sum_{1 \le j \le K} \|u\|_{L^\infty(B_{\lambda_n^{-1}/6}(x_j))}^2 \times \mu_n^\epsilon(B_{\lambda_n^{-1}/6}(x_j)) \\ & \lesssim_{d,N} \sum_{1 \le j \le K} \lambda_n^{d-2\epsilon N} \times \lambda^{ d \epsilon} \lesssim_{d} \lambda_n^{(\beta + 2 d\epsilon) + (d-2\epsilon N)} \to 0.\end{align*} $$

Proof. Indeed, by the translation invariance of these norms in the Fourier space, we only need to consider trigonometric polynomials whose Fourier supports are within the $2r$ neighborhood of the origin. Such functions form a finite dimensional space. Therefore, there exist the maximum and the minimum of the continuous function

$$ \begin{align*} u \in \{ L^2 (\mathbb{T}^d) : \| u \|_{L^2} =1\} \mapsto \|u\|_{L^2(\mathop{}\!\mathrm{d} \mu)}^2. \end{align*} $$

Finally, by the assumption that $\operatorname {\mathrm {supp}} \mu $ is not contained in the support of any nonzero trigonometric polynomials we have that the minimum is nonzero, concluding the proof.

Proof of Theorem 7.1 for $d = 3$

By Lemma 7.4, it remains to establish the trace and observability estimates for eigenfunctions with $\operatorname {\mathrm {diam}} \operatorname {\mathrm {supp}} \widehat {u}> r$ for some sufficiently large r.

For any such $L^2$ -normalized eigenfunction $u \in E_\lambda $ , one has $\lambda \gtrsim r$ . By Lemma 3.6, write $\mathcal {S}^2_\lambda = \bigcup _\alpha \Omega _\alpha $ where each $\Omega _\alpha $ lives in an affine subspace of dimension $\le 2$ and $\mathrm {dist}(\Omega _\alpha ,\Omega _\beta )> C \lambda ^{2/4!}$ some $C>0$ whenever $\alpha \ne \beta $ . Then we invoke the decomposition formula (7.2). We firstly estimate the second summand. Let $j_\lambda = \max \{j \in \mathbb {N} : 2^j \le C \lambda ^{2/4!}\}$ . Then

$$ \begin{align*} \biggl| \sum_{\alpha\ne\beta} \sum_{k \in \Omega_\alpha} \sum_{\ell \in \Omega_\beta} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \biggr| & \le \sum_{|k-\ell|> C \lambda^{2/4!}} |\widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}| \le \sum_{j \ge j_\lambda} \sum_{2^j \le |k-\ell| \le 2^{j+1}} |\widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}| \\ & \lesssim \sum_{j \ge j_\lambda} \max_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| \sum_{2^j \le |k-\ell| \le 2^{j+1}} \bigl( |\widehat{u}_k|^2 + |\widehat{u}_\ell|^2 \bigr) \\ & \lesssim \sum_{j \ge j_\lambda} N_3(\lambda,2^{j+1}) \max_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| \\ & \lesssim_\epsilon \sum_{j \ge j_\lambda} \lambda^{\epsilon/4!} 2^j \times 2^{-j(1+\epsilon)} \lesssim \sum_{j \ge j_\lambda} 2^{-j\epsilon/2} = o(1)_{r \to \infty}. \end{align*} $$

To conclude, we use the following claim: for each $\alpha $ , there holds

(7.3) $$ \begin{align} \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \sim_\mu \sum_{k \in \Omega_\alpha} |\widehat{u}_k|^2. \end{align} $$

Indeed, even though we are in the $d=3$ setting, the sets $(\Omega _\alpha )_\alpha $ each lies within a two dimensional affine hyperplane and is thus contained in a circle of radius $r_\alpha $ . Therefore the exact same proof as Theorem 4.1 (for the Fourier decay case) applies. This proof establishes the trace inequality and the semiclassical observability, that is, the observability inequality holds when $r_\alpha $ is sufficiently large. The full observability is guaranteed by Lemma 7.4.

Proof of Theorem 7.1 for $d\geq 3$

By Lemma 7.4, it remains to establish the trace and observability estimates for eigenfunctions with $\operatorname {\mathrm {diam}} \operatorname {\mathrm {supp}} \widehat {u}> r$ for some sufficiently large r.

We will achieve this by using mathematical induction on the affine dimension of $\operatorname {\mathrm {supp}} \widehat {u}$ . Precisely, we will prove the following statement: if r is sufficiently large, then for all $n \in \{1,\ldots ,d\}$ and any toral eigenfunction u satisfying $\operatorname {\mathrm {diam}} \operatorname {\mathrm {supp}} \widehat {u}> r$ and $\dim _{\mathrm {aff}} \operatorname {\mathrm {supp}} \widehat {u} = n$ , there holds

(7.4) $$ \begin{align} \|u\|_{L^2 (\mathop{}\!\mathrm{d} \mu)} \sim_{d,n,\mu,r} \| u \|_{L^2 (\mathbb{T}^d)}. \end{align} $$

We use mathematical induction. First we prove the case $n=1$ . In this case $\# \operatorname {\mathrm {supp}} \widehat {u} \le 2$ and we denote $\operatorname {\mathrm {supp}} \widehat {u} = \{ k, \ell \}$ . Let $\delta>0$ be such that $\sup _{\xi \in \mathbb {Z}^d \setminus 0} |\widehat {\mu } (\xi )| \leq 1 - \delta $ (Lemma 3.7). Then, as in Section 4,

$$ \begin{align*} \|u\|_{L^2}^2 \gtrsim \int_{\mathbb{T}^d} |u|^2 \mathop{}\!\mathrm{d} \mu = |\widehat{u}_k|^2 + |\widehat{u}_\ell|^2 + 2 \widehat{\mu}_{k-\ell} \widehat{u}_k \widehat{u}_\ell \geq \delta ( |\widehat{u}_k|^2 + |\widehat{u}_\ell|^2 ) = \delta \|u\|_{L^2}^2. \end{align*} $$

We now prove the induction step $n \Rightarrow n+1$ with $n \leq d-1$ . Fix any $L^2$ -normalized eigenfunction $u \in E_\lambda $ with $\dim _{\mathrm {aff}} \operatorname {\mathrm {supp}} \widehat {u} = n+1$ . Now $\operatorname {\mathrm {supp}} \widehat {u}$ is contained in an n-sphere of radius $\rho \gtrsim r$ . By Lemma 3.6, we write $\operatorname {\mathrm {supp}} \widehat {u} = \bigcup _\alpha \Omega _\alpha $ where $\dim _{\mathrm {aff}} \Omega _\alpha \le n$ for all $\alpha $ and $\mathrm {dist}(\Omega _\alpha ,\Omega _\beta ) \ge C_n \rho ^{2/(n+1)!}$ for some $C_n> 0$ whenever $\alpha \ne \beta $ . Then we invoke the decomposition formula (7.2) and for each $\alpha $ , we use the induction hypothesis and obtain

$$ \begin{align*} \sum_\alpha \sum_{k,\ell \in \Omega_\alpha} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \sim_{d,n,\delta} \sum_\alpha \sum_{k \in \Omega_\alpha} |\widehat{u}_k|^2 = \|u\|_{L^2}^2. \end{align*} $$

To estimate the second summand term in (7.2), we let $j_s = \max \{j \in \mathbb {N} : 2^j \le C_n s^{2/d!}\}$ and write

$$ \begin{align*} \sum_{|k-\ell|> c_{d-1} \lambda^{2/d!}} |\widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}| & \le \sum_{j \ge j_\rho} \sum_{2^j \le |k-\ell| \le 2^{j+1}} |\widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}| \\ & \lesssim \sum_{j \ge j_\rho} \max_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| \sum_{2^j \le |k-\ell| \le 2^{j+1}} \bigl( |\widehat{u}_k|^2 + |\widehat{u}_\ell|^2 \bigr) \\ & \lesssim \sum_{j \ge j_\rho} \sup_{k \in \operatorname{\mathrm{supp}} \widehat{u}} \# \bigl(\operatorname{\mathrm{supp}} \widehat{u} \cap B_{2^{j+1}}(k)\bigr) \times \max_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi|. \end{align*} $$

Next, we separate the discussion of the two cases: $n=d-1$ and $n<d-1$ :

• • If $n=d-1$ , then $\# \bigl (\operatorname {\mathrm {supp}} \widehat {u} \cap B_{2^{j+1}}(k)\bigr ) \le N_d(\lambda ,2^{j+1}) \lesssim _d \lambda ^\epsilon 2^{j(d-2)}$ and $\lambda =\rho \ge r$ . Therefore, the error term is bounded by

  $$ \begin{align*} \sum_{j \ge j_\lambda} \lambda^{ \epsilon/d!} 2^{j(d-2)} \times 2^{-j(d-2+\epsilon)} \lesssim \sum_{j \ge j_\lambda} 2^{-j\epsilon/2} = \sum_{j \ge j_r} 2^{-j\epsilon/2} = o(1)_{r \to \infty}. \end{align*} $$

• • If $n \leq d-2$ , then $\# \bigl (\operatorname {\mathrm {supp}} \widehat {u} \cap B_{2^{j+1}}(k)\bigr ) \lesssim _n 2^{n j} \lesssim 2^{(d-2) j}$ by the area method. Therefore, the error term is bounded by

  $$ \begin{align*} \sum_{j \ge j_\rho} 2^{(d-2)j} \times 2^{-j(d-2+\epsilon)} \lesssim \sum_{j \ge j_\rho} 2^{-j\epsilon} = \sum_{j \ge j_r} 2^{-j\epsilon} = o(1)_{r \to \infty}. \end{align*} $$

We finish the induction process by combining these estimates above and formula (7.2). They yield the following estimate:

$$ \begin{align*} \sum_{k,\ell \in \mathcal{S}^{d-1}_\lambda} \widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \sim_{d,n,\delta} 1 + o(1)_{r \to \infty}.\\[-49pt] \end{align*} $$

Proof. The proof follows the same outline as before, with only minor adjustments noted below:

• • The condition (7.5) implies the Fourier decay $|\widehat {\mu }_\xi | = o(|\xi |^{-(d-2)})_{\xi \to \infty }$ . Therefore we still have $\dim _{\mathrm {H}} \operatorname {\mathrm {supp}} \mu> d-1$ to apply Lemma 7.4.

• • In the $n=d-1$ case, we use the finer estimate $N_d(\lambda ,2^{j+1}) \lesssim _{d,\epsilon } \lambda ^{-1} 2^{j (d-1)} + \lambda ^{\epsilon /d!} 2^{j(d-3+\epsilon )}$ . Since it suffices to sum over those j with $ { \lambda ^{2/d!} } \lesssim 2^j \lesssim \lambda $ , we may further bound $N_d(\lambda ,2^{j+1}) \lesssim 2^{j(d-2)}$ . Therefore, the error term is bounded by

  $$ \begin{align*} \sum_{j \ge j_\lambda} 2^{j(d-2)} \sup_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| \le \sum_{j \ge j_r} 2^{j(d-2)} \max_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| = o(1)_{r \to \infty}. \end{align*} $$

• • In the $n\leq d-2$ case, it suffices to notice that, by keeping $\max _{2^j \le |\xi | \le 2^{j+1}} |\widehat {\mu }_\xi |$ in the summation, the remainder term satisfies the same estimate:

  $$ \begin{align*} \sum_{j \ge j_r} 2^{j(d-2)} \sup_{2^j \le |\xi| \le 2^{j+1}} |\widehat{\mu}_\xi| = o(1)_{r \to \infty}.\\[-43pt] \end{align*} $$

Proof of Theorem 8.1 for $d=3$

The proof follows the same structure as that of Theorem 7.1, specifically the proof for the $d=3$ case. That is, we apply the cluster decomposition and invoke the formula (7.2). We now estimate the second summand in (7.2). We first observe that, by Lemma 3.5, for all $\epsilon> 0$ it holds

$$ \begin{align*} \sum_{|k- \ell|> C \lambda^{2/4!}}|\widehat{\mu}_{k-\ell}|^2 & \leq \sum_{|\xi| > C \lambda^{2/4!}} \# \Bigl( (-\xi + \mathcal{S}^2_\lambda) \cap \mathcal{S}^2_\lambda \Bigr) \times |\widehat{\mu}_{\xi}|^2 \\ & \lesssim_\epsilon \lambda^{\epsilon/4!} \sum_{|\xi| > C \lambda^{2/4!}} |\widehat{\mu}_{\xi}|^2 \lesssim \sum_{|\xi| > C \lambda^{2/4!}} |\xi|^{2\epsilon} |\widehat{\mu}_{\xi}|^2. \end{align*} $$

Hence, by the Cauchy Schwarz inequality and using $g \in H^\epsilon $ ,

$$ \begin{align*} \sum_{|k- \ell|> C \lambda^{2/4!}} |\widehat{\mu}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}| & \leq \biggl ( \sum_{|k- \ell| > C \lambda^{2/4!}} |\widehat{\mu}_{k-\ell}|^2 \biggr )^{1/2} \biggl (\sum_{|k- \ell| > C \lambda^{2/4!}} | \widehat{u}_k \overline{\widehat{u}_\ell}|^2 \biggr )^{1/2} \\ & \lesssim_\epsilon \biggl ( \sum_{|\xi| > C \lambda^{2/4!}} |\xi|^{2 \epsilon} | \widehat{\mu}_{\xi}|^2 \biggr )^{1/2} \| u \|_{L^2}^2 = o(1)_{\lambda \to \infty}. \end{align*} $$

Then for each $\alpha $ , we obtain the estimate (7.3) by the exact same proof as in Theorem 4.1 (the case with $L^2$ densities). Particularly, the constant for the trace inequality is bounded by $\|g\|_{H^\epsilon }$ .

Proof of Theorem 8.1 for $d \geq 3$

It remains to establish (7.4) for any toral eigenfunction u satisfying $\operatorname {\mathrm {diam}} \operatorname {\mathrm {supp}} \widehat {u}> r$ , where r is sufficiently large, and $\dim _{\mathrm {aff}} \operatorname {\mathrm {supp}} \widehat {u} = n$ . Regarding the uniform trace inequality, we will also show that (8.1) for such eigenfunctions.

We use mathematical induction. The case $n=1$ is the same as in the proof of Theorem 7.1. We now prove the inductive step $n \Rightarrow n+1$ with $n \leq d-1$ . We perform the cluster decomposition on the n-sphere that supports $\widehat {u}$ , where u is an $L^2$ -normalized eigenfunction. Let $\rho $ be the radius of this sphere, then $\rho \gtrsim r$ . Recall that these clusters are well-separated: if $\alpha \ne \beta $ , then $\mathrm {dist}(\Omega _\alpha ,\Omega _\beta )> N = C_n \rho ^{2/(n+2)!} $ . Let $\mathbb {P}_N$ be a smooth spectral projector onto the frequency set $\{k \in \mathbb {Z}^d : |k| \le N\}$ , and write

(8.2) $$ \begin{align} \int_{\mathbb{T}^d} |u|^2 g \mathop{}\!\mathrm{d} x = \int_{\mathbb{T}^d} |u|^2 \mathbb{P}_N g \mathop{}\!\mathrm{d} x + \int_{\mathbb{T}^d} |u|^2 (1-\mathbb{P}_N) g \mathop{}\!\mathrm{d} x. \end{align} $$

To bound the first term on the right-hand side of (8.2), we use

$$ \begin{align*} \int_{\mathbb{T}^d} |u|^2 { \mathbb{P}_N g} \mathop{}\!\mathrm{d} x = \sum_{\alpha} \sum_{k, \ell \in \Omega_{\alpha}} \widehat{\mathbb{P}_N g}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell}. \end{align*} $$

Since $\Omega _\alpha $ is contained in an affine n-plane, we may apply the inductive hypothesis for each $\alpha $

$$ \begin{align*} \sum_{k \in \Omega_\alpha} |\widehat{u}_k|^2 \lesssim_{d,n,\delta} \sum_{k, \ell \in \Omega_{\alpha}} \widehat{\mathbb{P}_N g}_{k-\ell} \widehat{u}_k \overline{\widehat{u}_\ell} \lesssim_{d,n, \delta} \|\mathbb{P}_N g\|_{W^{\epsilon,\frac{d+1}{2}}} \sum_{k \in \Omega_\alpha} |\widehat{u}_k|^2. \end{align*} $$

We now bound the second term in the right-hand side of (8.2). We first observe that for $s \ge 0$ and $p\in [1,\infty )$ , there holds

$$ \begin{align*} \|\mathbb{P}_N\|_{W^{s,p} \to W^{s,p}} \lesssim_{d,s,p} 1, \quad \| (1- \mathbb{P}_N)g \|_{W^{s,p} \to L^p} \lesssim_{d,s,p} N^{-s}. \end{align*} $$

Using this, the Hölder’s inequality, the decoupling theorem [Reference Bourgain and Demeter9, Theorem 2.2] we conclude

$$ \begin{align*} \int_{\mathbb{T}^d} |u |^2 (1-\mathbb{P}_N) (g) \mathop{}\!\mathrm{d} x & \le \| u \|_{L^{ \frac{d+1}{d-1}}}^2 \| (1-\mathbb{P}_N) g \|_{ L^{\frac{d+1}{2}}} \lesssim_{d,\epsilon} N^{\epsilon/2} \| u \|_{L^2}^2 \| (1-\mathbb{P}_N)g \|_{ L^{\frac{d+1}{2}}} \\ & \lesssim_{d,\epsilon} N^{- \epsilon/2} \| u \|_{L^2}^2 \| (1-\mathbb{P}_N)g \|_{ W^{\epsilon, \frac{d+1}{2}}} \lesssim \| u \|_{L^2}^2 \| g \|_{ W^{\epsilon, \frac{d+1}{2}}} = o(1)_{r \to \infty}.\end{align*} $$