发表机构
University of Rochester（罗切斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 $\mathbb{Z}_N^d$ 上带复杂性参数的渐近谱综合，通过傅里叶比刻画集中指数，证明有限尺度综合估计并确定精确综合常数，揭示了素数幂与素数乘积下端点行为的差异。
AI 中文摘要
我们利用傅里叶比（即酉傅里叶变换的计数测度 $\ell^1$ 范数与 $\ell^2$ 范数之比）研究 $\mathbb{Z}_N^d$ 上的渐近谱综合。对于基数为 $N^{\alpha+o(1)}$ 的支集，傅里叶比集中指数满足 $0\leq\kappa\leq\frac{\alpha}{2}$。我们证明了一个有限尺度综合估计，该估计使用函数的傅里叶比或其支集指示函数的傅里叶比。特别地，指示函数条件统一适用于支集上的任意权重。当 $2\leq p<\frac{2(d-2\kappa)}{\alpha-2\kappa}$ 时，一致的 $\ell^p$ 界蕴含最大傅里叶系数的衰减，且在最大集中时允许每个有限 $p\geq2$。陪集的随机并表明，在素数幂模数下，对于每个规定环境维数中的每个允许对，有限临界指数是精确的。我们还确定了具有传递仿射对称群的集合的精确综合常数。对于抛物面，该常数是一个显式的除数求和。该公式证明了沿固定奇素数幂的端点综合，尽管端点沿素数失败。沿不同小素数的乘积，综合扩展到更大的指数。这些族具有相同的傅里叶比指数。最后，我们将傅里叶集中与广义 Salem 估计、加性能量、傅里叶代数范数以及扩展估计的必要条件联系起来。
英文摘要
We study asymptotic spectral synthesis on $\mathbb Z_N^d$ using the Fourier ratio, the quotient of the counting-measure $\ell^1$ and $\ell^2$ norms of the unitary Fourier transform. For supports of cardinality $N^{α+o(1)}$, the Fourier-ratio concentration exponent satisfies $0\leqκ\leq\fracα{2}$. We prove a finite-scale synthesis estimate that uses either the Fourier ratio of a function or that of its support indicator. In particular, the indicator condition applies uniformly to arbitrary weights on the support. Uniform $\ell^p$ bounds imply decay of the largest Fourier coefficient when $2\leq p<\frac{2(d-2κ)}{α-2κ}$, with every finite $p\geq2$ allowed at maximal concentration. Random unions of cosets show that the finite critical exponent is sharp for every admissible pair in each prescribed ambient dimension, along prime-power moduli. We also determine the exact synthesis constant for sets with a transitive group of affine symmetries. For paraboloids this constant is an explicit divisor sum. The formula proves endpoint synthesis along powers of a fixed odd prime, although the endpoint fails along primes. Along products of distinct small primes, synthesis extends to a larger exponent. These families have the same Fourier-ratio exponent. Finally, we relate Fourier concentration to generalized Salem estimates, additive energy, Fourier algebra norms, and necessary conditions for extension estimates.
