Spectral synthesis with the complexity parameter in ℤNd
Abstract
We study asymptotic spectral synthesis on ℤNd using the Fourier ratio, the quotient of the counting-measure ℓ1 and ℓ2 norms of the unitary Fourier transform. For supports of cardinality Nα+o(1), the Fourier-ratio concentration exponent satisfies 0≤κ≤α/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 ℓp bounds imply decay of the largest Fourier coefficient when 2≤ p<(2(d−2κ))/(α−2κ), with every finite p≥2 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.
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BibTeX
@misc{iosevich2026spectral,
title = {{Spectral synthesis with the complexity parameter in $\mathbbZ_N^d$}},
author = {Alex Iosevich and Zhihe Li and Ke Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25404},
eprint = {2609.25404},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25404},
}
RIS
TY - UNPB
TI - Spectral synthesis with the complexity parameter in ℤ_N^d
AU - Alex Iosevich
AU - Zhihe Li
AU - Ke Yu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25404
UR - https://arxiv.org/abs/2609.25404
AB - We study asymptotic spectral synthesis on ℤ_N^d using the Fourier ratio, the quotient of the counting-measure ℓ^1 and ℓ^2 norms of the unitary Fourier transform. For supports of cardinality N^α+o(1), the Fourier-ratio concentration exponent satisfies 0≤κ≤α/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 ℓ^p bounds imply decay of the largest Fourier coefficient when 2≤ p<(2(d−2κ))/(α−2κ), with every finite p≥2 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.
ER -