Openai 2026 geometric case erdos similarity conjecture

../

proposition_2_1: The manuscript's main construction: for a fixed ratio q and every p in (0,1), an open 1-periodic subset of the line of density at most 6p that meets every translate of t{q^n : n >= 1}, t in [1,2]; Theorem 1.1 follows by a summable union of its dyadic dilations and reflections.

theorem_1_1: The manuscript's main claim: for each fixed ratio q in (0,1) and each eta in (0,1), a compact set in [0,1] of measure above 1-eta that contains no translated, nontrivially dilated copy of the geometric progression q^n, for either sign of the dilation; the geometric-progression case of Problem 120.

OpenAI, The geometric case of the Erdős similarity conjecture, OpenAI Math Release preprint, October 5, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026; the held PDF, geometric-erdos-similarity.pdf in the release, is retained as openai_2026_geometric_case_erdos_similarity_conjecture.pdf, and the release's TeX bundle in the same folder is the TeX source cited below.

@misc{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026,
  author = {{OpenAI}},
  title = {{The geometric case of the Erd\H{o}s similarity conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf}{OAI:The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026}},
  year = {2026}
}

Attestation as the release states it. The release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all of them have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block, and adds no sentence about how this manuscript was produced or checked; the manuscript's text names no human author and no verification step. These are the source's historical attestations, not this corpus's review. No refereed publication, no arXiv version and no independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalogue (lean/formalization.yaml) lists no formalization for this manuscript. The family's Lean page (lean/docs/084.md), linked from the release's contents map, describes a formalization of the companion dyadic manuscript named below and none of this one, with the comparator statement file lean/ComparatorChallenges/DyadicAvoidance.lean; this was read statically from the release's family page lean/docs/084.md and its comparator file, not built, replayed or audited for fidelity in this repository, and it covers no ratio other than $1/2$. Whether a release declaration settles the problem is recorded on the problem's claim pages, not on this card.

Companions. The release files this manuscript in one family with The dyadic case of the Erdős similarity conjecture (its card), which treats the ratio $q=1/2$ alone; the present manuscript states (Section

1. that its theorem at $q=1/2$ gives that dyadic case, so the companion is the special case and this manuscript the general-ratio claim.

Read status: claims checked for Theorem 1.1 and Proposition 2.1, and for the statements of Lemmas 3.1--3.3, 4.1--4.3 and 5.1, read clause by clause in the TeX source (sections/01-introduction.tex lines 17--24, sections/02-periodic.tex lines 12--19, sections/03-windows.tex, sections/04-routing.tex, sections/05-scales.tex) on 2026-10-07, against the held PDF for page numbers; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The manuscript is thirteen pages: five sections and a fourteen-entry reference list. main.tex inputs sections/01-introduction.tex (which inputs sections/01-history.tex), 02-periodic.tex, 03-windows.tex, 04-routing.tex and 05-scales.tex; figures/window-block.tex is the one figure.

• Section 1, Introduction (pp. 1--3). Defines a set $A\subseteq\mathbb R$ to be measure universal when every Lebesgue-measurable set of positive measure contains a nontrivial affine copy $x+sA$ ($s\ne0$), names the Erdős similarity conjecture, cited to Erdős's 1974 Mathematica Balkanica problem list (Problem 4.33.7*), as the assertion that no infinite set is measure universal, writes $G_q=\{q^n:n\ge1\}$ and states Theorem 1.1: for every $q\in(0,1)$ and $\eta\in(0,1)$ a compact $E_{q,\eta}\subseteq[0,1]$ of measure above $1-\eta$ meets no $x+sG_q$ with $s\ne0$. The set may depend on $q$; $q=1/2$ gives the dyadic case; the conjecture for arbitrary infinite sets is called a separate question. Subsection 1.1, Background and related work, records that finite sets are universal (continuity of translation in $L^1$), the Falconer and Eigen theorem for sequences with $a_{n+1}/a_n\to1$, the Humke--Laczkovich covering characterization, Kolountzakis's probabilistic criterion, and Chlebík's translation-invariant criterion (a bounded infinite set is nonuniversal if it has arbitrarily large finite subsets whose normalized minimum gap has negative logarithm $o(m)$), with a two-line check that geometric progressions fail that criterion (normalized gap at most $q^{m-2}/(1-q)$); then the additive results (Bourgain's three-sum theorem, Kolountzakis's double sums including $G_{1/2}+G_{1/2}$, the 2026 Mora Cuellar--Iosevich--Kulkarni--Rojas Aravena--Yavicoli theorem that adding to or subtracting from an arbitrary infinite set a geometric null sequence gives a nonuniversal set), the Rajchman-measure result of the same group (which excludes countable sets), the Cruz--Lai--Pramanik dimension-one avoiding sets (of measure zero) and the Feng--Lai--Xiong bi-Lipschitz embedding theorem (so the restriction to affine maps matters). The manuscript places its proof in Kolountzakis's probabilistic approach, names Chlebík (Section 5) and Kolountzakis--Papageorgiou (Section 3.1) as precedents for its random cells, its discretization of scales at a fixed center and its integration of the exceptional-center probabilities, describes the Solymosi and Tom USRA reports as incomplete dyadic random-cell constructions, and states its own contribution as the finite routing construction with local control of the scale count. Subsection 1.2, The proof mechanism, is a prose overview of Sections 2--5.
• Section 2, A periodic hitting set and the global deduction (pp. 3--4). Fixes $q$, writes $\rho(A)=m(A\cap[0,1))$ for a $1$-periodic set, and states Proposition 2.1: for every $p\in(0,1)$ an open $1$-periodic $H$ with $\rho(H)\le6p$ meets $x+tG_q$ for every real $x$ and every $t\in[1,2]$. Proves Theorem 1.1 from it: with $p_k=\eta 4^{-|k|}/64$ and $H_k$ from the proposition, the open set $C=\bigcup_{k\in\mathbb Z}(2^kH_k\cup-2^kH_k)$ has $m(C\cap[0,1])\le7\eta/16$, $E_{q,\eta}=[0,1]\setminus C$ is compact, and writing $s=\pm2^kt$ with $t\in[1,2)$ reduces every signed dilation to the normalized one. The dyadic factors only normalize $s$ and need no relation between $2$ and $q$.
• Section 3, Grids, preorder windows, and stable centers (pp. 4--7). With $c=4/(1-q)$ and $N_b=2^{\lceil\log_2(cq^{-b})\rceil}$, so $cq^{-b}\le N_b<2cq^{-b}$, defines the nested dyadic grids $\Gamma_b=N_b^{-1}\mathbb Z$ and the periodic keys $J_b(z)=\lfloor N_b\{z\}\rfloor$ (cells closed on the left). Takes a complete ordered $M$-ary tree of height $d$ with $K$ edges, orders the edges in preorder, and assigns each edge $e$ an index window $W_e=[a_e,b_e]\cap\mathbb N$ of length $r_h$ depending on the height $h$ of its parent, with gaps of exactly $g$ indices between consecutive windows, $r_1=r_0$, $r_h=\max\{r_0,g+\sigma_{h-1}\}$ where $\sigma_h$ is the span of a height-$h$ subtree, and first index $n_0$ with $2q^{n_0}\le1/4$; $\mathcal N$ is the union of the windows. Lemma 3.1: the span of an edge's window together with its child subtree is at most $2r_h$ (Figure 1). Defines stable centers (no grid point of the predecessor window's grid in $(x,x+2q^{a_e}]$ for any noninitial window) and proves Lemma 3.2: the unstable centers have density at most $4Kcq^{g+1}$, and at a stable center a translation by $tq^n$, $n\in W_e$, $t\in[1,2]$, leaves every key of an earlier edge unchanged. Lemma 3.3: at any center the center and its $r_h$ translated points have pairwise distinct keys at resolution $b_e$ and finer.
• Section 4, Random routing and independent tests (pp. 7--10). Attaches to each nondefault edge a table of independent fair bits indexed by grid cells, and to each leaf a table of independent Bernoulli-$p$ bits; routes each point from the root to the first child whose selector bit is one, defaulting to the last child, and puts the point in the random periodic set $B$ when its leaf's terminal bit is one. Lemma 4.1: $\mathbb E\rho(B)=p$. Exposing a center's addressed entry in every selector table gives a sigma-field $\mathcal F_x$; the route of $x$ has no default choice with probability $(1-2^{1-M})^d$. Lemma 4.2: at a stable center whose first default vertex is $U$, every translated point from the windows of $U$'s nondefault edges reaches $U$ and rejects the earlier children, so a successful local test $Q_i$ (selector on edge $i$ times the terminal bit after routing from child $i$) is a hit in $B$. Lemma 4.3: conditional on an exposure atom and a fixed $t\in[1,2]$, the $(M-1)r_h$ local tests fail together with probability exactly $(1-p/2)^{(M-1)r_h}$, after showing the selector entries and the terminal addresses are distinct.
• Section 5, All normalized scales and all centers (pp. 10--12). Lemma 5.1: with $D_{M,q}(r)=4+2(M-1)r(1+2cq^{-2r})$, the conditional probability that some $t\in[1,2]$ misses $B$ at every index of $\mathcal N$ is at most $D_{M,q}(r)e^{-p(M-1)r/2}$, by listing the finitely many $t$ at which a translated point crosses the finest grid of the edge-and-subtree block (Lemma 3.1 bounds that grid) and applying Lemma 4.3 at each representative; the manuscript credits the finite boundary-representative idea to Chlebík (proof of Theorem 15) and Kolountzakis--Papageorgiou (Section 3.1). Proof of Proposition 2.1: choose $M$ with $p(M-1)/2>2\log(1/q)$, $d$ with $(1-2^{1-M})^d<p$, $g$ with $4Kcq^{g+1}<p$ and $r_0=r_*$ with $D_{M,q}(r)e^{-p(M-1)r/2}<p$ for $r\ge r_*$; then a stable center misses some normalized scale with probability at most $2p$. Enlarge $B$ to an open periodic $B^+$ with $\rho(B^+)\le\rho(B)+p$, define the closed periodic set $R$ of centers still missed at some $t\in[1,2]$ by the indices in $\mathcal N$ (closed as a projection from a compact product), get $\mathbb E\rho(R)\le3p$, pick an outcome with $\rho(B^+)+\rho(R)\le5p$, cover $R$ by an open periodic $V$ with $\rho(V)\le\rho(R)+p$ (outer regularity), and set $H=B^+\cup V$; a center in $R$ is hit because $tq^n\to0$ puts late terms inside the open neighborhood $V$ (the open-neighborhood repair is credited to Tom, Lemma 0.2, and Chlebík). Those late indices need not lie in $\mathcal N$, and the proposition allows every $n\ge1$.
• References (pp. 12--13): Erdős 1974; Falconer 1984; Eigen 1985; Bourgain 1987; Kolountzakis 1997; Humke--Laczkovich 1998; Chlebík 2015 (arXiv preprint); Solymosi 2011 and Tom 2015 (UBC USRA reports); Kolountzakis--Papageorgiou 2025; Mora Cuellar et al. 2026 and Iosevich et al. 2026 (arXiv preprints); Cruz--Lai--Pramanik 2023; Feng--Lai--Xiong 2024.

The proof is presented as self-contained: it rests on finite product probability spaces, the nesting of dyadic grids, and elementary measure theory on the circle (outer regularity, projection of a closed set from a compact product). No cited theorem is used as a premise; the citations in Sections 1 and 5 are precedents for the method. The manuscript flags nothing as numerical, computer-assisted or conditional, and the release folder holds no verification material beyond the PDF, its build files and the README.

Bears on

• Problem 120: claimed partial answer. The problem asks whether every infinite $A\subseteq\mathbb R$ is avoided, up to a nontrivial affine copy $aA+b$, by some set of positive measure; Theorem 1.1 claims this for $A=\{q^n:n\ge1\}$ with any fixed $q\in(0,1)$, with the avoiding set compact in $[0,1]$ and of measure as close to $1$ as desired, and with both signs of $a$ covered. The avoiding set depends on $q$, and the manuscript says nothing about any infinite set that is not a geometric progression, so the general question stays as the page records it. The claim is unverified here; the page's status rests on its acceptance evidence, not on this card.
• Jung, Lai and Mooroogen (2024): the survey's status line records the conjecture as open for exponentially decaying sequences such as $2^{-n}$; Theorem 1.1 is a later claim covering the geometric progressions within that case, one fixed ratio at a time, unverified here, and the survey's broader class of exponentially decaying sequences is not addressed; the survey's Theorem 1.3 (Falconer, Eigen) is the complementary slow-decay case the manuscript cites as background.
• Erdős (1978): that survey states the similarity conjecture (p. 123) for every infinite set on the line; Theorem 1.1 is a claimed instance of it for geometric progressions. The manuscript cites the 1974 Mathematica Balkanica statement rather than this one; the relation is the conjecture's statement, not a cited input.