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Statement

Let $m$ be Lebesgue measure on $\mathbb R$ and, for $q\in(0,1)$, let $G_q=\{q^n:n\in\mathbb N,\ n\ge1\}$. A nontrivial affine copy of a set $A$ is $x+sA$ with $x\in\mathbb R$ and $s\in\mathbb R\setminus\{0\}$.

Theorem 1.1. For every $q\in(0,1)$ and every $\eta\in(0,1)$ there is a compact set $E_{q,\eta}\subseteq[0,1]$ with $m(E_{q,\eta})>1-\eta$ such that for every $x\in\mathbb R$ and every $s\in\mathbb R\setminus\{0\}$ some integer $n\ge1$ has

$$x+sq^n\notin E_{q,\eta}.$$

The quantifiers are as the manuscript prints them: the ratio $q$ and the measure deficit $\eta$ are fixed first, the set then depends on both, and the conclusion runs over every real translation and every nonzero dilation of either sign. The manuscript adds (Section 1, p. 1) that the set "may depend on $q$", that the case $q=1/2$ yields compact subsets of $[0,1]$ of measure arbitrarily close to $1$ containing no affine copy of $\{2^{-n}:n\ge1\}$ under a dilation of either sign, and that the conjecture for arbitrary infinite sets "is a separate question". The abstract (p. 1) says the result "makes no simultaneous assertion for different ratios".

Source. OpenAI, The geometric case of the Erdős similarity conjecture, release folder preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026; TeX sections/01-introduction.tex, environment thm:main (lines 17--24), PDF p. 1; proof from Proposition 2.1 in sections/02-periodic.tex lines 26--70, PDF pp. 3--4; read. The card records the provenance and attestations.

Read depth. Claims checked: the statement, the definitions of measure universality and $G_q$, and the three qualifying sentences above were read clause by clause in the TeX source and located in the PDF. The deduction from Proposition 2.1 and the five-section proof of that proposition were read for their structure only (below); no step was checked. Nothing here is independently reviewed.

Proof pointer

Section 2 reduces the theorem to Proposition 2.1, an open $1$-periodic set $H$ of density at most $6p$ that meets $x+tG_q$ for every real $x$ and every normalized dilation $t\in[1,2]$. Given the proposition, the deduction is short: for each $k\in\mathbb Z$ take $H_k$ with $p_k=\eta4^{-|k|}/64$, let $C=\bigcup_k(2^kH_k\cup(-2^kH_k))$ and $E_{q,\eta}=[0,1]\setminus C$. The set $C$ is open and symmetric, so $E_{q,\eta}$ is compact; each $\pm2^kH_k$ is $2^k$-periodic and has measure $2^k\rho(H_k)$ in each period, so $m(C\cap[0,1])\le12\sum_kp_k\max(1,2^k)=7\eta/16<\eta$. For $s>0$ write $s=2^kt$ with $t\in[1,2)$ and apply the proposition to $H_k$ at the center $2^{-k}x$; for $s<0$ reflect. The dyadic factors normalize $s$ only and use no relation between $2$ and $q$.

The manuscript proves Proposition 2.1 in Sections 3--5 by a random construction: nested dyadic grids of resolution comparable to $q^{-b}/(1-q)$ at index $b$, a finite ordered $M$-ary tree whose edges carry consecutive index windows in preorder with gaps between them, random selector and terminal tables routing each point to a leaf, independence of the local tests at a stable center's first default vertex (Lemma 4.3), a finite set of scale representatives whose count is controlled by one window length (Lemma 5.1), and an open-neighborhood repair of the closed set of exceptional centers using $tq^n\to0$. The hypothesis $q<1$ enters through the grid separation (Lemma 3.3), the growth $q^{-2r}$ against the decay $e^{-p(M-1)r/2}$ in the choice of $M$, and the repair step. The hypothesis $\eta<1$ only fixes the measure budget.

Dependencies

None at statement level. The proof is presented as self-contained, using finite product probability spaces, nesting of dyadic grids, and elementary measure theory on the circle (outer regularity; a projection from a compact product is closed). The works cited in Sections 1 and 5 (Kolountzakis 1997; Chlebík 2015; Kolountzakis and Papageorgiou 2025; Tom 2015) are named as precedents for the method, not invoked as premises. None was checked here.

Bears on

• Problem 120: claimed partial answer, the case $A=\{q^n:n\ge1\}$ for each fixed $q\in(0,1)$, with the avoiding set compact in $[0,1]$, of measure above $1-\eta$, and avoiding dilations of both signs. The general question for an arbitrary infinite $A$ is not addressed. The claim is unverified here; the page's status rests on its acceptance evidence.