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Statement

For a measurable $1$-periodic set $A\subseteq\mathbb R$ let $\rho(A)=m(A\cap[0,1])$, the measure of $A$ over one period; the manuscript at times treats such a set as a subset of the circle $\mathbb T=\mathbb R/\mathbb Z$. Let $D=\{2^{-n}:n\ge1\}$.

Lemma 2.1 (Periodic hitting sets). For every $p\in(0,1)$ there is an open $1$-periodic set $H\subseteq\mathbb R$ with $\rho(H)\le6p$ such that

$$\forall x\in\mathbb R\ \ \forall t\in[1,2]\ \ \exists n\in\mathbb Z_{\ge1}: \qquad x+t2^{-n}\in H.$$

A single $H$ is uniform in both the center $x$ and the normalized scale $t$; only the hitting index $n$ is allowed to vary with them.

Source. OpenAI, The dyadic case of the Erdős similarity conjecture, release folder preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026; TeX sections/periodic.tex, environment lem:periodic (lines 13--20), PDF p. 4; proof spread over sections/windows.tex, sections/routing.tex, sections/tests.tex, sections/scales.tex and sections/repair.tex (Sections 3--5, PDF pp. 4--14), completed in sections/repair.tex lines 13--130 (PDF pp. 13--14); read. The card records the provenance and attestations.

Read depth. Claims checked: the statement and the definition of $\rho$ were read clause by clause in the TeX source and located in the PDF, together with the statements of the intermediate results Lemma 3.1, Lemmas 4.1--4.3, Lemma 5.1 and Proposition 5.2. The proofs were read for their structure only (below); no step was checked. Nothing here is independently reviewed.

Proof pointer

The proof fixes $p$ and runs through five deterministic choices, a random construction, and a repair.

• Section 3 fixes the combinatorics. Integers $M\ge2$ and $d\ge1$ with $p(M-1)/2\ge10$ and $(1-2^{1-M})^d<p$ give a complete ordered $M$-ary tree of height $d$ with $K$ edges; a gap $g$ with $K2^{2-g}<p$; a threshold $r_*$ with $20Mr(1+2^{2r+2})\exp(-p(M-1)r/2)<p$ for all $r\ge r_*$; and window lengths $r_h$ by a bottom-up recursion so that a child block's whole index span is at most $2r_h$ (display (5)). Each edge $e$ receives a block $W_e$ of consecutive dyadic indices in edge preorder, starting at $3$, with $g$ unused indices between blocks; $\mathcal N$ is their union. Periodic grid keys $J_b(z)=\lfloor2^{b+2}\{z\}\rfloor$ are nested. A center $x$ is stable when no translate indexed by a later window crosses a boundary of the preceding window's grid; Lemma 3.1 shows the stable set $G$ has $\rho(G^c)<p$ and that for $x\in G$ every translate $x+t2^{-n}$, $n\in W_e$, $t\in[1,2]$, keeps every earlier window's key.
• Section 4 builds the random set. Each nondefault edge $e=(P,i)$, $i<M$, carries a fair random table $S_e$ indexed by the keys at resolution $b_e$ (the window's last index); each leaf carries a Bernoulli-$p$ table $T_L$ at its incoming window's resolution. A point is routed from the root by the first child whose selector reads $1$, the last child by default, and $B$ is the set of points whose terminal entry reads $1$; $\mathbb E\rho(B)=p$. Lemma 4.1: a fixed center's route never takes a default child with probability $(1-2^{1-M})^d<p$. For a stable center whose route first defaults at node $U$ of height $h$, Lemma 4.2 shows that the translates indexed by the window of child $i$ of $U$ repeat the center's route to $U$ and its rejections of children $1,\dots,i-1$, so a local predicate $Q_i$ (selector on $(U,i)$ times the terminal entry reached from $U_i$) equal to $1$ forces the translate into $B$. Lemma 4.3: at a fixed $t\in[1,2]$, conditional on the center's exposed selector entries, the $(M-1)r_h$ tested selector and terminal addresses are pairwise distinct, so all local tests fail with probability exactly $(1-p/2)^{(M-1)r_h}$.
• Section 5 passes to all scales and all centers. Lemma 5.1: the local predicates depend on the key at resolution $b_i^*$ (the block's largest endpoint), so as $t$ runs over $[1,2]$ each tested translate crosses at most $1+2^{2r_h+2}$ grid boundaries, and a set of at most $20Mr_h(1+2^{2r_h+2})$ representative scales reproduces every pattern of predicate values. Proposition 5.2 combines Lemmas 4.1--4.3 and 5.1 by a union bound over representatives: $\mathbb P(\exists t\in[1,2]\ \forall n\in\mathcal N:\ x+t2^{-n}\notin B)\le2p$ for every stable $x$. Subsection 5.1 completes the lemma: enlarge each outcome's $B_\omega$ to an open periodic $B_\omega^+$ adding at most $p$ to its density; the exceptional-center set $R_\omega$ (centers with some $t\in[1,2]$ missing $B_\omega^+$ at every $n\in\mathcal N$) is closed and periodic, by compactness of $\mathbb T\times[1,2]$, with $\mathbb E\rho(R_\omega)\le2p+\rho(G^c)\le3p$; one outcome has $\rho(B^+)+\rho(R)\le5p$; an open periodic $\varepsilon$-neighborhood $V$ of $R$ has $\rho(V)\le\rho(R)+p$, and $H=B^+\cup V$. A center outside $R$ is hit inside $\mathcal N$; a center in $R$ is hit by every $x+t2^{-n}$ with $2^{1-n}<\varepsilon$, which lies in $V$ because $x+t2^{-n}\to x$.

The hypothesis $p>0$ makes $M$ and $d$ exist; $p<1$ keeps $p$ a probability for the terminal tables. The restriction $t\in[1,2]$ is what makes the grid crossings per translate finite and is removed in Section 6 by dyadic rescaling.

Dependencies

None at statement level. The manuscript names Kolountzakis 1997 (random cells, scale discretization, open-cover repair), Chlebík 2015, Kolountzakis and Papageorgiou 2025 and the periodic blocking sets of Iosevich, Kulkarni, Mora Cuéllar, Rojas Aravena and Yavicoli 2026 as precedents for the method and says that all estimates needed are supplied in the text. None was checked here.

Bears on

• Problem 120: the lemma is the input from which the manuscript deduces the claimed dyadic case of the question (Theorem 1.1); on its own it says nothing about the problem, and the claim is unverified here. The page's status rests on its acceptance evidence.
• [[analysis/openai_2026_geometric_case_erdos_similarity_conjecture/proposition_2_1|The companion's Proposition 2.1]]: the geometric-case manuscript's periodic hitting statement for $\{q^n\}$ with general $q\in(0,1)$ plays the same role there as this lemma does here; comparison only, neither verified here.