Spectral synthesis with the complexity parameter in $\mathbb{Z}_N^d$

Abstract

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.