Let A be a countable and discrete subset of \({\mathbb R}^d\) , \(d \ge 2\) , of positive upper Beurling density. Let K denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known that \({\mathcal E}(A)=\{e^{2 \pi i x \cdot a}\}_{a \in A}\) cannot serve as an orthogonal basis for \(L^2(K)\) (Iosevich et al., Amer. J. Math. 123: 115–120, 2001). In this paper, we prove that even approximate average orthogonality is an obstacle to the existence of an exponential frame in the following sense. Let A be as above and \(\phi \ge 0\) be a continuous monotonically non-increasing function on \([0, \infty )\) such that the approximate orthogonality condition holds \( {\left ( \frac {1}{2^j} \int _{2^j}^{2^{j+1}} \phi ^p(t) dt \right )}^{\frac {1}{p}} \leq c_j 2^{-j\frac {d+1}{2}} \quad \text{and} \quad \ |\widehat {\chi }_K(a-a')| \leq \phi (\rho ^(a-a'))\ \ \forall a \neq a , a,a' \in A, \) where \(\rho ^{}\) is the Minkowski functional on \(K^{*}\) , the dual body of K. Then, if \(\limsup _{j \to \infty } c_j=0,\) then the upper density of A is equal to 0, hence \({\mathcal E}(A)={\{e^{2 \pi i x \cdot a} \}}_{a \in A}\) is not a frame for \(L^2(K)\) . The case \(p=\infty \) was previously established by the authors of this paper in Iosevich and Mayeli (On complete and incomplete exponential systems. Proceeding of AMS). The point is that if \({\mathcal E}(A)\) is a frame for \(L^2(K)\) , then very few pairs of distinct exponentials \(e^{2 \pi i x \cdot a}\) , \(e^{2 \pi i x \cdot a'}\) from \({\mathcal E}(A)\) come anywhere near being orthogonal. Our proof uses a generalization of Bourgain’s result on pinned distances determined by sets of positive Lebesgue upper density in \({\mathbb R}^d\) , \(d \ge 2\) . We also improve the \(L^{\infty }\) version of this result originally established in Iosevich and Mayeli (On complete and incomplete exponential systems. Proceeding of AMS). By using an extension of the combinatorial idea from Iosevich and Rudnev (Int. Math. Res. Not. 50:2671–26852003), we prove that under the \(L^{\infty }\) hypothesis, A is finite if \(d \neq 1 \mod 4\) . If \(d=1\) mod 4, A may be infinite, but if it is, then it must be a subset of a line.

Approximate Orthogonality, Bourgain’s Pinned Distance Theorem, and Exponential Frames

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发表日期：

2026年

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Alex Iosevich,
Azita Mayeli

摘要

Let A be a countable and discrete subset of \({\mathbb R}^d\) , \(d \ge 2\) , of positive upper Beurling density. Let K denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known that \({\mathcal E}(A)=\{e^{2 \pi i x \cdot a}\}_{a \in A}\) cannot serve as an orthogonal basis for \(L^2(K)\) (Iosevich et al., Amer. J. Math. 123: 115–120, 2001). In this paper, we prove that even approximate average orthogonality is an obstacle to the existence of an exponential frame in the following sense. Let A be as above and \(\phi \ge 0\) be a continuous monotonically non-increasing function on \([0, \infty )\) such that the approximate orthogonality condition holds \( {\left ( \frac {1}{2^j} \int _{2^j}^{2^{j+1}} \phi ^p(t) dt \right )}^{\frac {1}{p}} \leq c_j 2^{-j\frac {d+1}{2}} \quad \text{and} \quad \ |\widehat {\chi }_K(a-a')| \leq \phi (\rho ^*(a-a'))\ \ \forall a \neq a , a,a' \in A, \) where \(\rho ^{*}\) is the Minkowski functional on \(K^{*}\) , the dual body of K. Then, if \(\limsup _{j \to \infty } c_j=0,\) then the upper density of A is equal to 0, hence \({\mathcal E}(A)={\{e^{2 \pi i x \cdot a} \}}_{a \in A}\) is not a frame for \(L^2(K)\) . The case \(p=\infty \) was previously established by the authors of this paper in Iosevich and Mayeli (On complete and incomplete exponential systems. Proceeding of AMS). The point is that if \({\mathcal E}(A)\) is a frame for \(L^2(K)\) , then very few pairs of distinct exponentials \(e^{2 \pi i x \cdot a}\) , \(e^{2 \pi i x \cdot a'}\) from \({\mathcal E}(A)\) come anywhere near being orthogonal. Our proof uses a generalization of Bourgain’s result on pinned distances determined by sets of positive Lebesgue upper density in \({\mathbb R}^d\) , \(d \ge 2\) . We also improve the \(L^{\infty }\) version of this result originally established in Iosevich and Mayeli (On complete and incomplete exponential systems. Proceeding of AMS). By using an extension of the combinatorial idea from Iosevich and Rudnev (Int. Math. Res. Not. 50:2671–26852003), we prove that under the \(L^{\infty }\) hypothesis, A is finite if \(d \neq 1 \mod 4\) . If \(d=1\) mod 4, A may be infinite, but if it is, then it must be a subset of a line.