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The Erdős similarity problem for null sequences with adjacent ratios bounded below

  • Kan Jiang
  • Lifeng Xi
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Abstract

Let (an) be a positive null sequence with liminfn→∞an+1/an>0. For every η>0, we construct a compact set E⊂[0,1] with Lebesgue measure greater than 1−η containing no nontrivial affine copy of {an:n≥1}. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set for any prescribed countable family of sequences, with infinitely many distinct points omitted from every affine copy.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026erdős,
  title = {{The Erdős similarity problem for null sequences with adjacent ratios bounded below}},
  author = {Kan Jiang and Lifeng Xi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11449},
  eprint = {2610.11449},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11449},
}
RIS
TY  - UNPB
TI  - The Erdős similarity problem for null sequences with adjacent ratios bounded below
AU  - Kan Jiang
AU  - Lifeng Xi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11449
UR  - https://arxiv.org/abs/2610.11449
AB  - Let (a_n) be a positive null sequence with liminf_n→∞a_n+1/a_n>0. For every η>0, we construct a compact set E⊂[0,1] with Lebesgue measure greater than 1−η containing no nontrivial affine copy of {a_n:n≥1}. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set for any prescribed countable family of sequences, with infinitely many distinct points omitted from every affine copy.
ER  -