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arXiv · 2610.11449

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

Kan Jiang·Lifeng Xi
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Abstract

Let $(a_n)$ be a positive null sequence with $\liminf_{n\to\infty}a_{n+1}/a_n>0$. For every $η>0$, we construct a compact set $E\subset[0,1]$ with Lebesgue measure greater than $1-η$ containing no nontrivial affine copy of $\{a_n:n\ge1\}$. 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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Kan Jiang, Lifeng Xi. 2026-10-08. The Erdős similarity problem for null sequences with adjacent ratios bounded below. https://arxiv.org/abs/2610.11449

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