Openai 2026 dyadic case erdos similarity conjecture

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lemma_2_1: The periodic hitting lemma that carries the manuscript's construction: for every p in (0,1) an open 1-periodic subset of the line of density at most 6p that meets x+t{2^{-n}} for every real center x and every normalized dilation t in [1,2]; proved in Sections 3-5 by random routing on a finite tree, and rescaled dyadically in Section 6 to give Theorem 1.1.

theorem_1_1: The manuscript's main claim: for every eta in (0,1) a compact set in [0,1] of measure above 1-eta that contains no translated, nontrivially dilated copy of the dyadic sequence 2^{-n}, for either sign of the dilation; the dyadic case of Problem 120, deduced from the periodic hitting lemma.

OpenAI, The dyadic case of the Erdős similarity conjecture, OpenAI Math Release preprint, September 25, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_dyadic_case_erdos_similarity_conjecture.pdf, and the release's TeX bundle in the same folder is the TeX source cited below.

@misc{OAI:The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{The dyadic case of the Erd\H{o}s similarity conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf}{OAI:The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026}},
  year = {2026}
}

Attestation, as the release states it. The release's root README says the 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 adds nothing beyond the title, the author line "OpenAI", the date and the citation block; the TeX source names no human author and carries no statement on how the text was produced. These are the source's own attestations, recorded here as history, not as 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.

Formalization, as the release lists it. The release's Lean page (lean/docs/084.md) names this manuscript as its accompanying paper and describes the formalized statement as the dyadic case: for every $0<\eta<1$ a compact $E\subset[0,1]$ of measure greater than $1-\eta$ such that for every real $x$ and every real $s\ne0$, positive or negative, some $x+s2^{-n}$ lies outside $E$. The comparator statement file it names is lean/ComparatorChallenges/DyadicAvoidance.lean, theorem OAI.Problem310.dyadic_affine_avoidance (an internal label of the release; it does not refer to Erdős Problem 310), whose companion configuration points to the solution module OAI/MeasureTheory/DyadicAvoidance/Main.lean and permits the three standard axioms. The release's formalization catalogue lean/formalization.yaml at the held revision carries no entry for this manuscript or this comparator, so the Lean page and the catalogue disagree on whether the result is listed. All of this is read statically from the release's catalogue; not built, replayed or audited for fidelity in this repository. A Lean statement about the sequence $\{2^{-n}\}$ is not a proof of the Erdős problem, which asks about every infinite set.

Companion. The release files this manuscript in a family with The geometric case of the Erdős similarity conjecture (October 5, 2026), whose card is [[analysis/openai_2026_geometric_case_erdos_similarity_conjecture/_index|the geometric-case card]]; that manuscript claims the same conclusion for $\{q^n:n\ge1\}$ with every fixed ratio $q\in(0,1)$; its introduction notes that $q=1/2$ yields the dyadic case and that for general $q$ it keeps nested dyadic grids with resolutions depending on $q$, but it does not cite this manuscript. The present manuscript is thus the $q=1/2$ case of the companion's claim and the only member of the family that the release's Lean page (lean/docs/084.md) lists as formalized.

Read status: claims checked for Theorem 1.1 (the main theorem) and Lemma 2.1 (the periodic hitting lemma), read clause by clause in the TeX source (sections/introduction.tex lines 24--33, sections/periodic.tex lines 13--20) on 2026-10-07, together with the statements of Lemma 3.1, Lemmas 4.1--4.3, Lemma 5.1 and Proposition 5.2; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF has sixteen pages; the TeX source is paper.tex with eight files under sections/ for six sections; Sections 4 and 5 each span two files. Theorems, lemmas and propositions share one counter per section.

• Section 1, Introduction (sections/introduction.tex, pp. 1--3). Defines a nontrivial affine copy $x+sA$ ($s\ne0$) and measure universality, cites the conjecture to Erdős's 1974 Mathematica Balkanica problem list (Problem 4.33.7*; the 1978 survey restatement held at Erdős's 1978 survey is not cited), notes that every finite set is measure universal by translation continuity in $L^1$, fixes $D=\{2^{-n}:n\ge1\}$ and states Theorem 1.1. The text says the theorem treats the single pattern $D$ with all its signed affine copies and leaves the conjecture for general infinite sets open. Subsection 1.1 surveys related results: Falconer 1984 and Eigen 1985 (sequences with $a_{n+1}/a_n\to1$), Humke and Laczkovich 1998, Kolountzakis 1997 (finite-gap criterion, and sums such as $D+D$), Chlebík 2015, Bourgain 1987 (sums of three infinite sets), two 2026 arXiv preprints of Mora Cuéllar, Iosevich, Kulkarni, Rojas Aravena and Yavicoli (a geometric sequence plus an arbitrary infinite set; a Rajchman-measure criterion that cannot apply to a countable set), Cruz, Lai and Pramanik 2023 (dimension-one avoiding sets of measure zero), Feng, Lai and Xiong 2024 (bi-Lipschitz embedding) and the withdrawn 2020 Cruz--Lai--Pramanik preprint. It checks that $D$ fails the Kolountzakis and Chlebík gap criteria. Subsection 1.2 outlines the construction and names its precedents (random cells and scale discretization in Kolountzakis 1997, Chlebík 2015 and Kolountzakis--Papageorgiou 2025; an open-cover repair of exceptional centers in Kolountzakis 1997; an incomplete dyadic attempt in a 2011 undergraduate report). It states that the manuscript itself supplies every estimate the proof uses.
• Section 2, The periodic hitting problem (sections/periodic.tex, pp. 3--4). Defines the density $\rho(A)=m(A\cap[0,1])$ of a $1$-periodic set and states Lemma 2.1: for every $p\in(0,1)$ an open $1$-periodic $H$ with $\rho(H)\le6p$ that meets $x+tD$ for every $x\in\mathbb R$ and every $t\in[1,2]$. Sections 3--5 prove it.
• Section 3, Separated index windows (sections/windows.tex, pp. 4--7). Fixes a complete ordered $M$-ary tree of height $d$ with $p(M-1)/2\ge10$ and $(1-2^{1-M})^d<p$, a gap $g$ with $K2^{2-g}<p$ ($K$ the number of edges), a threshold $r_*$ making $20Mr(1+2^{2r+2})\exp(-p(M-1)r/2)<p$ for all $r\ge r_*$, and window lengths $r_h$ by a bottom-up recursion. Each edge receives a block $W_e$ of dyadic indices in preorder with $g$ unused indices between blocks; $\mathcal N$ is their union, and display (5) bounds a child block's span by $2r_h$. Subsection 3.2 defines the periodic grid keys $J_b(z)=\lfloor2^{b+2}\{z\}\rfloor$ and the stable centers $G$, and Lemma 3.1 proves $\rho(G^c)<p$ and that for a stable center the translates $x+t2^{-n}$, $n$ in a window, keep every earlier window's key.
• Section 4, Random routing and independent tests (sections/routing.tex and sections/tests.tex, pp. 7--11). Attaches a fair random selector table to every nondefault edge and a Bernoulli-$p$ terminal table to every leaf, routes each point to a leaf by the first successful selector (default child when all fail) and defines the random periodic set $B$ with $\mathbb E\rho(B)=p$. Lemma 4.1: a fixed center's route never takes a default child with probability $(1-2^{1-M})^d<p$. Lemma 4.2: for a stable center and its first default node $U$, the translates indexed by the window of child $i$ follow the same route to $U$ and reject children $1,\dots,i-1$, so a local predicate $Q_i=1$ forces membership in $B$. Lemma 4.3: at a fixed scale $t\in[1,2]$, conditional on the exposed selector entries at the center, all $(M-1)r$ local tests fail with probability exactly $(1-p/2)^{(M-1)r}$, after showing the tested addresses are distinct.
• Section 5, All normalized scales and all centers (sections/scales.tex and sections/repair.tex, pp. 11--14). Lemma 5.1 discretizes the scale: a finite set of at most $20Mr(1+2^{2r+2})$ representative scales reproduces every pattern of local predicate values as $t$ varies in $[1,2]$. Proposition 5.2 combines Lemmas 4.1--4.3 and 5.1 by a union bound into $\mathbb P(\exists t\in[1,2]\ \forall n\in\mathcal N:\ x+t2^{-n}\notin B)\le2p$ for every stable center. Subsection 5.1 completes the proof of Lemma 2.1: enlarge each realization $B_\omega$ to an open periodic $B_\omega^+$ adding at most $p$ in density, show the exceptional-center set $R_\omega$ is closed and periodic with $\mathbb E\rho(R_\omega)\le3p$, pick one outcome with $\rho(B^+)+\rho(R)\le5p$, and cover $R$ by an open periodic neighborhood $V$ of density at most $\rho(R)+p$ so that the tail of $x+tD$ enters $V$; $H=B^+\cup V$.
• Section 6, The compact avoiding set (sections/global.tex, pp. 14--15). Proves Theorem 1.1 from Lemma 2.1 with $p_j=\eta2^{-j}/24$: the open set $C=\bigcup_j(2^{-j}H_j\cup-2^{-j}H_j)$ has $m(C\cap[0,1])\le\eta/2$, and $E=[0,1]\setminus C$ is compact; a positive scale $s$ is normalized to $t=2^ks\in[1,2)$ with $j=\max\{1,k\}$, and a negative scale is reflected.
• References (references.bib, pp. 15--16): fourteen entries, listed in Section 1 above.

The manuscript flags nothing as numerical, computer-assisted or conditional; the probability space is finite, one outcome is chosen by an averaging argument, and the proof invokes no external theorem beyond standard measure theory and compactness. The release provides no verification/ folder for this manuscript.

Bears on

• Problem 120: Theorem 1.1 is a claimed answer to the exact question for the single set $A=\{2^{-n}:n\ge1\}$, which the page's Statement quantifies over every infinite $A$; the manuscript itself says the general conjecture is not addressed. The claim is unverified here, and the page's status rests on acceptance evidence, not on this card.
• [[analysis/jung_2024_fifty_years_erdos_similarity_conjecture/_index|Jung, Lai and Mooroogen's survey]]: its status item, a 2024 record, lists the conjecture as open for exponentially decaying sequences such as $2^{-n}$; Theorem 1.1 is a claimed settlement of exactly that case, and acceptance evidence for the manuscript, which this card does not supply, decides between the two.