PreprintCC BY 4.0arXiv
Remarks on the Geometry of Sets in ℤd with Small Fourier L1 Norm
Abstract
We study geometric inverse problems for finite sets A⊂ ℤd whose Fourier L1 norm, or Wiener norm, |A|:=∫𝕋d|∑a∈ Ae(a· t)|dt is subpolynomial in |A|. In particular, we show a variety of geometric phenomena are incompatible with small Fourier L1 norm. Our principal application concerns spherical Freiman models. Suppose |A|=|A|o(1) and D⊆ A is Freiman isomorphic to a set S of lattice points on a sphere; then, |D|/|A| must be polynomially small in |A|. Moreover, if |D|/|A| is not too small, almost all of S lies in a small number of spherical caps, and almost all caps have rich additive structure. The proof uses a new mechanism for studying additive structure across subsets of A, along with decoupling results of Bourgain-Demeter.
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BibTeX
@misc{burgin2026remarks,
title = {{Remarks on the Geometry of Sets in $\mathbbZ^d$ with Small Fourier $L^1$ Norm}},
author = {Alex Burgin and Junzhe Mao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26619},
eprint = {2609.26619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26619},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on the Geometry of Sets in ℤ^d with Small Fourier L^1 Norm
AU - Alex Burgin
AU - Junzhe Mao
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.26619
UR - https://arxiv.org/abs/2609.26619
AB - We study geometric inverse problems for finite sets A⊂ ℤ^d whose Fourier L^1 norm, or Wiener norm, |A|:=∫_𝕋^d|∑_a∈ Ae(a· t)|dt is subpolynomial in |A|. In particular, we show a variety of geometric phenomena are incompatible with small Fourier L^1 norm. Our principal application concerns spherical Freiman models. Suppose |A|=|A|^o(1) and D⊆ A is Freiman isomorphic to a set S of lattice points on a sphere; then, |D|/|A| must be polynomially small in |A|. Moreover, if |D|/|A| is not too small, almost all of S lies in a small number of spherical caps, and almost all caps have rich additive structure. The proof uses a new mechanism for studying additive structure across subsets of A, along with decoupling results of Bourgain-Demeter.
ER -