{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25404",
  "author_total": 3,
  "authors": [
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Zhihe Li"
    },
    {
      "name": "Ke Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fnx9ar62enm8m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25404",
  "fulltext_url": null,
  "id": "fnx9ar62enm8m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25404",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214127088",
  "pdf_url": "https://arxiv.org/pdf/2609.25404",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fnx9ar62enm8m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25404",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25404"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral synthesis with the complexity parameter in $\\mathbb{Z}_N^d$",
  "title_text": "Spectral synthesis with the complexity parameter in ℤ_N^d",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fnx9ar62enm8m",
  "venue": "arXiv",
  "versions": []
}