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The Erdős similarity conjecture and Rajchman measures

  • A. Iosevich
  • N. Kulkarni
  • N. Mora Cuéllar
  • I. Rojas Aravena
  • A. Yavicoli
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Abstract

Let A⊆ℝ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every ε∈(0,1), there is a closed, $1$-periodic, nowhere dense set E⊆ℝ such that m(E∩ I)≥1−ε for every interval I of length $1$, while E contains no affine copy of A. Thus every set supporting a Rajchman measure satisfies the Erdős similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Ivašev-Musatov, showing that for every Hausdorff gauge h there is an h-null compact Rajchman support K satisfying dim̅Blog K=dimPlog K=1. The value $1$ is sharp for both dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{iosevich2026erdős,
  title = {{The Erdős similarity conjecture and Rajchman measures}},
  author = {A. Iosevich and N. Kulkarni and N. Mora Cuéllar and I. Rojas Aravena and A. Yavicoli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04456},
  eprint = {2609.04456},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04456},
}
RIS
TY  - UNPB
TI  - The Erdős similarity conjecture and Rajchman measures
AU  - A. Iosevich
AU  - N. Kulkarni
AU  - N. Mora Cuéllar
AU  - I. Rojas Aravena
AU  - A. Yavicoli
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04456
UR  - https://arxiv.org/abs/2609.04456
AB  - Let A⊆ℝ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every ε∈(0,1), there is a closed, $1$-periodic, nowhere dense set E⊆ℝ such that m(E∩ I)≥1−ε for every interval I of length $1$, while E contains no affine copy of A. Thus every set supporting a Rajchman measure satisfies the Erdős similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Ivašev-Musatov, showing that for every Hausdorff gauge h there is an h-null compact Rajchman support K satisfying dim̅_B^log K=dim_P^log K=1. The value $1$ is sharp for both dimensions.
ER  -