==== TABLE OF CONTENTS ==== The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/routing.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/windows.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//paper.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/global.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/introduction.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/periodic.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/repair.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/routing.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/scales.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/tests.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/windows.tex The-dyadic-case-of-the-Erdos-similarity-conjecture//references.bib =========================== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/routing.tex ==== \begin{figure}[!htb] \centering \begin{tikzpicture}[x=1cm,y=1cm,font=\small,>=Latex] \node[draw,rounded corners,fill=gray!8,minimum width=4.3cm] (prefix) at (0,3.15) {common choices at strict ancestors}; \node[draw,circle,inner sep=3pt] (u) at (0,1.85) {$U$}; \draw[->,thick] (prefix)--node[right] {stable earlier keys}(u); \node[draw,rounded corners,fill=blue!8] (childi) at (-3,.15) {$U_i$, $i,thick,blue!70!black] (u)--node[above left,align=right] {$y_n(t)$: selector $i$ succeeds\\earlier children rejected}(childi); \draw[->,thick,orange!80!black] (u)--node[above right,align=left] {$x$: all $M-1$ selectors vanish\\default choice}(childm); \node[align=center,below=5mm of childi] (terminal) {local route to $L_i(y_n(t))$\\terminal entry equals $1$}; \draw[->,blue!70!black] (childi)--(terminal); \end{tikzpicture} \caption{Route preservation for $x\in G$, $n\in W_{(U,i)}$ and $t\in[1,2]$, conditional on $Q_i(y_n(t))=1$. The center $x$ and its translate $y_n(t)$ make the same required choices above the first default node $U$. At $U$, the center takes child $M$, while the translate rejects the earlier children and takes child $i$. The local terminal success then places $y_n(t)$ in $B$. The diagram is conditional on the successful local predicate; it does not assume independence of the chosen leaves.} \label{fig:routing} \end{figure} ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/routing.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/windows.tex ==== \begin{figure}[htbp] \centering \begin{tikzpicture}[x=1cm,y=1cm,font=\small] \node[anchor=west] at (0,2.5) {Child blocks in edge preorder}; \foreach \start/\index in {0/1,4.5/2,9.4/M}{ \fill[blue!10] (\start,1.6) rectangle (\start+1.35,2.2); \fill[gray!12] (\start+1.35,1.6) rectangle (\start+3.95,2.2); \draw (\start,1.6) rectangle (\start+3.95,2.2); \draw (\start+1.35,1.6)--(\start+1.35,2.2); \node at (\start+.675,1.9) {$W_{(P,\index)}$}; \node at (\start+2.65,1.9) {edges below $P_{\index}$}; } \draw[->] (3.98,1.9)--(4.43,1.9); \node at (8.89,1.9) {$\cdots$}; \node[anchor=west] at (0,.95) {One block, expanded for $h\ge2$}; \fill[blue!10] (1,.05) rectangle (7.1,.65); \fill[gray!12] (7.7,.05) rectangle (12.35,.65); \draw (1,.05) rectangle (7.1,.65); \draw (7.7,.05) rectangle (12.35,.65); \node at (4.05,.35) {$W_e$: length $r_h$}; \node at (7.4,.35) {$g$}; \node at (10.025,.35) {below $P_i$: span $\sigma_{h-1}$}; \node[anchor=north] at (1,0) {$a_e$}; \node[anchor=north] at (7.1,0) {$b_e$}; \node[anchor=north] at (12.35,0) {$b_e^*$}; \draw (1,-.65)--(12.35,-.65); \draw (1,-.55)--(1,-.75); \draw (12.35,-.55)--(12.35,-.75); \node[anchor=north] at (6.675,-.7) {$b_e^*-a_e+1=r_h+g+\sigma_{h-1}\le2r_h$}; \end{tikzpicture} \caption{Each edge window precedes the windows below its child, and the child blocks are listed in order. The lower diagram includes the gap before a nonempty child subtree. An edge into a leaf has no following subtree or internal gap. Lengths are schematic.} \label{fig:windows} \end{figure} ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//figures/windows.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//paper.tex ==== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{microtype} \usepackage{booktabs} \def\DoNotLoadEpstopdf{} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing} \usepackage[unicode,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \usepackage{bookmark} \pdfinfoomitdate=1 \pdftrailerid{} \pdfsuppressptexinfo=15 \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\T}{\mathbb T} \newcommand{\E}{\mathbb E} \newcommand{\Pbb}{\mathbb P} \newcommand{\N}{\mathbb N} \hypersetup{pdftitle={The dyadic case of the Erdos similarity conjecture},pdfauthor={OpenAI}} \title{The dyadic case of the Erd\H{o}s similarity conjecture} \author{OpenAI} \date{September 25, 2026} \begin{document} \maketitle \begin{abstract} We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence $\{2^{-n}:n\ge1\}$. The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erd\H{o}s similarity conjecture. \end{abstract} \tableofcontents \input{sections/introduction} \input{sections/periodic} \input{sections/windows} \input{sections/routing} \input{sections/tests} \input{sections/scales} \input{sections/repair} \input{sections/global} \bibliographystyle{plain} \bibliography{references} \end{document} ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//paper.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/global.tex ==== \section{The compact avoiding set}\label{sec:global} The periodic hitting sets cover scales in $[1,2]$. We now combine their dyadic dilations, with summable densities, to cover every nonzero scale while removing arbitrarily little measure from $[0,1]$. \begin{proof}[Proof of Theorem~\ref{thm:main}] Fix $\eta\in(0,1)$ and put \[ p_j=\frac{\eta}{24}\,2^{-j}\qquad(j=1,2,\ldots). \] Each $p_j$ belongs to $(0,1)$. By Lemma~\ref{lem:periodic}, choose an open $1$-periodic set $H_j\subset\R$ with $\rho(H_j)\le6p_j$ such that \begin{equation}\label{global:normalized-hits} \forall y\in\R\ \forall t\in[1,2]\ \exists n\in\mathbb Z_{\ge1}: \qquad y+t2^{-n}\in H_j. \end{equation} All these sets are fixed before any center or scale is chosen. Define \[ C=\bigcup_{j\ge1}\bigl(2^{-j}H_j\cup(-2^{-j}H_j)\bigr), \qquad E=[0,1]\setminus C. \] Dilations and reflections preserve openness, so $C$ is open. Its paired definition gives $-C=C$. Thus $E$ is a closed subset of $[0,1]$, and is compact. For each $j$, periodicity and a change of variables give the exact identities \begin{align*} m\bigl((2^{-j}H_j)\cap[0,1]\bigr) &=2^{-j}m\bigl(H_j\cap[0,2^j]\bigr)=\rho(H_j),\\ m\bigl((-2^{-j}H_j)\cap[0,1]\bigr) &=2^{-j}m\bigl(H_j\cap[-2^j,0]\bigr)=\rho(H_j). \end{align*} Indeed, each interval on the right contains exactly $2^j$ unit periods; their endpoints do not affect Lebesgue measure. Countable subadditivity therefore yields \[ m(C\cap[0,1])\le2\sum_{j\ge1}\rho(H_j) \le12\sum_{j\ge1}p_j=\frac{\eta}{2}. \] Consequently \[ m(E)=1-m(C\cap[0,1])\ge1-\frac{\eta}{2}>1-\eta. \] It remains to show that every signed affine copy of $D$ meets $C$. Let $x\in\R$ and first suppose that $s>0$. Choose an integer $k$ with \[ t=2^k s\in[1,2), \] and set $j=\max\{1,k\}$. Apply~\eqref{global:normalized-hits} to the center $2^j x$ and the normalized scale $t$. There is an integer $n\ge1$ such that \[ 2^j x+t2^{-n}\in H_j. \] Multiplying by $2^{-j}$ gives \[ x+s2^{-(n+j-k)}\in2^{-j}H_j\subset C. \] The index $N=n+j-k$ is an integer and satisfies $N\ge n\ge1$, because $j\ge k$. Hence this point belongs to the original copy $x+sD$. The argument applies to every real center and every positive scale, including arbitrarily small scales, for which larger values of $j$ are available. If $s<0$, apply the positive-scale conclusion to the center $-x$ and scale $-s$. For some integer $N\ge1$, \[ -(x+s2^{-N})=(-x)+(-s)2^{-N}\in C. \] Since $-C=C$, it follows that $x+s2^{-N}\in C$ as well. In either case, the copy $x+sD$ has a point in $C$, which is disjoint from $E$. Thus the fixed compact set $E$ excludes every such copy, as required. \end{proof} ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/global.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/introduction.tex ==== \section{Introduction}\label{sec:introduction} Let $m$ denote Lebesgue measure on $\R$. For a set $A\subseteq\R$, a \emph{nontrivial affine copy} of $A$ is a set \[ x+sA=\{x+sa:a\in A\},\qquad x\in\R,\quad s\in\R\setminus\{0\}. \] The set $A$ is \emph{measure universal} if every Lebesgue-measurable subset of $\R$ of positive measure contains such a copy. The Erd\H{o}s similarity conjecture asserts that no infinite subset of $\R$ is measure universal; the problem was posed by Erd\H{o}s in \cite[Problem~4.33.7*]{Erdos1974Problems}. Every nonempty finite pattern $F$ is measure universal. Indeed, if a measurable set contains a subset $E_0$ of finite positive measure, translation continuity in $L^1$ gives $m(\bigcap_{a\in F}(E_0-sa))\to m(E_0)>0$ as $s\to0$. A point in this intersection supplies an affine copy of $F$. The difficulty therefore lies in keeping infinitely many points in the same positive-measure set. We consider the single sequence \[ D=\{2^{-n}:n\in\N,\ n\ge1\}. \] \begin{theorem}\label{thm:main} For every $\eta\in(0,1)$, there is a compact set $E_\eta\subseteq[0,1]$ with $m(E_\eta)>1-\eta$ such that \[ \text{for every }x\in\R\text{ and every }s\in\R\setminus\{0\}, \quad x+sD\nsubseteq E_\eta. \] Equivalently, for every such $x,s$ there is an integer $n\ge1$ with $x+s2^{-n}\notin E_\eta$. \end{theorem} Thus the dyadic sequence is not measure universal. The theorem concerns one infinite pattern and all of its signed affine copies; the conjecture for arbitrary infinite sets is not addressed. Compactness and the near-full-measure conclusion follow directly from the open sets used in the construction. \subsection{Related results} Classical results concern slowly decreasing sequences. Falconer~\cite{Falconer1984} and Eigen~\cite{Eigen1985} proved nonuniversality for positive sequences $a_n\downarrow0$ with $a_{n+1}/a_n\to1$; see also the proof in \cite[Corollary~1]{HumkeLaczkovich1998}. Humke and Laczkovich give a finite hitting-set characterization and further relative-gap criteria \cite[Theorems~1--2]{HumkeLaczkovich1998}. Kolountzakis's finite-gap criterion uses arbitrarily large finite subsets of a positive sequence: the minimum gaps, divided by their largest points, must have negative logarithms growing sublinearly in the cardinalities \cite[Theorem~3]{Kolountzakis1997}. Chleb\'ik gives a translation-invariant formulation normalized by the diameter \cite[Theorem~15]{Chlebik2015}. Dyadic subsets do not satisfy either condition: for $n\ge2$ dyadic points $a_1>\cdots>a_n$, the minimum gap is at most $a_{n-1}\le2^{-(n-2)}a_1$, whereas the diameter $a_1-a_n$ is at least $a_1/2$. Either normalized minimum gap is therefore at most $2^{-(n-3)}$. Additive structure gives another family of results. Bourgain proved that a sum of three infinite sets is nonuniversal \cite{Bourgain1987}, and Kolountzakis treated sums of two copies of certain sequences, including $D+D$ \cite[Section~3.4]{Kolountzakis1997}. The distinction between the dyadic sequence and larger patterns is important. Mora Cu\'ellar, Iosevich, Kulkarni, Rojas Aravena and Yavicoli prove nonuniversality for the sum or difference of a nonconstant geometric sequence tending to zero and an arbitrary infinite set \cite[Theorem~2]{MoraCuellarEtAl2026Twofold}. Their Introduction explicitly separates the single dyadic sequence from this two-fold sumset result. Avoiding a larger pattern does not imply avoidance of a subset of that pattern. Iosevich, Kulkarni, Mora Cu\'ellar, Rojas Aravena and Yavicoli prove a nonuniversality criterion for sets supporting a Rajchman probability measure, that is, a probability measure whose Fourier transform tends to zero at infinity \cite[Theorem~1.5]{IKMRY2026Rajchman}. A countable set cannot support such a measure \cite[Proposition~4.1]{IKMRY2026Rajchman}, so that criterion does not apply to $D$. Their more general uniform-density criterion on the circle \cite[Proposition~3.2]{IKMRY2026Rajchman} also fails for $D\cup\{0\}$: for every integer $k\ge1$, dilation by $2^k$ leaves its image modulo one unchanged, and that image stays a distance $1/4$ from $3/4$. Other notions of size and other transformation classes also lead to different questions. For certain sequences tending to zero, Cruz, Lai and Pramanik obtain sets of Hausdorff dimension one avoiding affine copies of the sequence together with its limit point; the avoiding sets in their construction have Lebesgue measure zero \cite[preprint, Corollary~1.2 and Proposition~4.4]{CruzLaiPramanik2023}. In the other direction, Feng, Lai and Xiong's bi-Lipschitz embedding theorem applies to the dyadic sequence \cite[preprint, Theorem~1.1]{FengLaiXiong2024}. Such nonlinear embeddings are compatible with the affine nonuniversality in Theorem~\ref{thm:main}. A separate earlier preprint of Cruz, Lai and Pramanik claiming the full similarity conjecture was withdrawn because of a gap in its Proposition~3.3 \cite{CruzLaiPramanik2020Withdrawn}. \subsection{The construction} Our main step is to construct an open, $1$-periodic set of small density that meets every $x+tD$ with $x\in\R$ and $t\in[1,2]$. The next section states this intermediate result precisely. Dyadic rescaling and reflection then produce a small open set meeting every signed affine copy of $D$; its complement in $[0,1]$ is the desired compact set. Periodic blocking sets and summable measure budgets also appear in \cite[Lemma~3.1 and Proposition~3.2]{IKMRY2026Rajchman}; the finite construction below supplies the required hitting property for $D$. Random cell constructions, discretization of the dilation parameter at a fixed center, and integration over centers occur in Kolountzakis \cite[Section~3.2]{Kolountzakis1997}, Chleb\'ik \cite[Section~5]{Chlebik2015}, and Kolountzakis--Papageorgiou \cite[Section~3.1]{KolountzakisPapageorgiou2025}. Exceptional centers are repaired by an open cover in Kolountzakis \cite[Sections~2.2 and~3.2]{Kolountzakis1997}; Chleb\'ik also describes adding the centers themselves when the pattern contains zero \cite[Section~3, final remark]{Chlebik2015}. We use the open-neighborhood form, since $0$ is a limit point of $D$ but is not in $D$. An earlier attempt to extend the random-cell approach to the dyadic sequence is described by Solymosi \cite[Section~3]{Solymosi2011Sparse}; that report does not claim a completed construction. The argument below implements these principles with separated index windows and a finite routing construction. All estimates needed for the proof are supplied here. To construct the periodic set, we first assign each point of the line to one leaf of a finite ordered tree with $M$ children at each internal node and $d$ levels. Independent fair binary tables, evaluated on dyadic cells, determine the route. At each node, the first successful test chooses a child; if all tests fail, the last child is chosen by default. Each leaf carries a second table whose entries select points with a small prescribed probability $p$. This keeps the expected density small. At a node, the default child is chosen only when all $M-1$ tested bits vanish. Although this is rare for large $M$, taking the depth $d$ sufficiently large makes a default somewhere along the center's route likely. At the first such node, all nondefault choices have been rejected by the center. Carefully chosen dyadic translates preserve the route to that node and its earlier rejections, but consult fresh entries at their current-window resolution. Many nondefault children and many translates then supply many potential hits. Two features make those tests useful simultaneously for every normalized scale. First, an interval of dyadic indices is assigned to each edge, in the order in which the tree is traversed. Gaps between these intervals preserve the earlier routing decisions for all but a small set of centers. Second, each interval is long enough to control every cell resolution in its child subtree. This bounds the number of different local tests as the scale varies. We expose the tables in stages, proving that the terminal entries used by these adaptively chosen routes are distinct before multiplying their failure probabilities. A finite union bound then controls all scales in $[1,2]$. The resulting set may still miss copies centered in a small closed set. An open neighborhood of those centers repairs every such copy, since $x+t2^{-n}$ converges to $x$. This last step is what changes an estimate on exceptional centers into a statement valid at every center. The windows and their spatial stability are constructed in Section~\ref{sec:windows}. Section~\ref{sec:routing} defines the random set and proves the independent-trial estimate. Section~\ref{sec:scales} handles the continuum of normalized scales and removes the exceptional centers. Section~\ref{sec:global} completes the signed-scale deduction of Theorem~\ref{thm:main}. ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/introduction.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/periodic.tex ==== \section{The periodic hitting problem}\label{sec:periodic} For a measurable $1$-periodic set $A\subseteq\R$, write \[ \rho(A)=m(A\cap[0,1]) \] for its density in one period. We will sometimes identify such a set with its image in the circle $\T=\R/\mathbb Z$, equipped with normalized Lebesgue measure. This identification preserves the density. Open or closed subsets of the circle lift to open or closed periodic subsets of $\R$. \begin{lemma}[Periodic hitting sets]\label{lem:periodic} For every $p\in(0,1)$, there exists an open $1$-periodic set $H\subseteq\R$ such that $\rho(H)\le6p$ and \[ \forall x\in\R\ \forall t\in[1,2]\ \exists n\in\mathbb Z_{\ge1}: \qquad x+t2^{-n}\in H. \] \end{lemma} The same set $H$ works for every center and every scale in the displayed interval. The witness $n$ may depend on both. In Section~\ref{sec:global}, we apply the lemma with a summable sequence of parameters, rescale the resulting periodic sets by powers of two, and include their reflections. This handles every nonzero real scale while keeping the total measure removed from $[0,1]$ small. For the proof of Lemma~\ref{lem:periodic}, fix $p\in(0,1)$. We construct a random periodic set $B$ with expected density $p$, together with a finite set $\mathcal N$ of positive integer indices. Outside a deterministic set of centers of density less than $p$, the probability that some $t\in[1,2]$ avoids $B$ at all indices in $\mathcal N$ will be at most $2p$. We then enlarge $B$ to an open set, bound the expected measure of its exceptional centers, and cover those centers by a small open set. The last step uses further indices in the dyadic tail; $\mathcal N$ need not provide the hits at the repaired centers. ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/periodic.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/repair.tex ==== \subsection{Removing exceptional centers} \label{subsec:repair} The pointwise probability bound will control the measure of the centers at which some normalized scale is missed. We first enlarge the random set to an open set, so that those exceptional centers form a closed set. A small open neighborhood of that closed set will then provide hits from the dyadic tail. This follows the open-cover completion of Kolountzakis \cite[Sections~2.2 and~3.2]{Kolountzakis1997}; we give the closedness, measure and tail arguments explicitly. \begin{proof}[Completion of the proof of Lemma~\ref{lem:periodic}] The probability space \(\Omega\) of all table entries is finite. For each outcome \(\omega\), the set \(B_\omega\) is a union of cells from one finite dyadic grid per period: its finest key determines every table lookup. Regard a periodic set as a subset of the circle \(\mathbb T=\mathbb R/\mathbb Z\), with normalized Lebesgue measure equal to the density of its periodic lift. Taking the closures of the selected cells adds only finitely many endpoints. Enlarging these closed arcs by sufficiently small open circular neighborhoods therefore gives an open periodic set \(B_\omega^+\supseteq B_\omega\) with \begin{equation} \rho(B_\omega^+)\le\rho(B_\omega)+p. \label{eq:open-enlargement} \end{equation} For example, if the grid has \(Q\) cells, enlargement by circular distance \(p/(4Q)\) adds at most \(p/2\) to the measure of their union. For the empty set use the empty enlargement. Since there are only finitely many outcomes, these choices introduce no measurable-selection issue. Equation~\eqref{eq:mean-density} gives \begin{equation} \mathbb E\rho(B_\omega^+)\le2p. \label{eq:open-mean} \end{equation} Define the exceptional-center set \begin{equation} R_\omega= \left\{x\in\mathbb R: \exists t\in[1,2]\ \forall n\in\mathcal N,\ x+t2^{-n}\notin B_\omega^+\right\}. \label{eq:exceptional-centers} \end{equation} This set is periodic and closed. To prove closedness without any endpoint convention on a fundamental interval, let \(\pi:\mathbb R\to\mathbb T\) be the quotient map and let \(B_{\omega,\mathbb T}^+\subset\mathbb T\) denote the open image of \(B_\omega^+\). Each map \[ (\pi x,t)\longmapsto\pi(x+t2^{-n}) \quad\hbox{from }\mathbb T\times[1,2]\hbox{ to }\mathbb T \] is continuous. The conditions that all these images avoid \(B_{\omega,\mathbb T}^+\), for \(n\in\mathcal N\), consequently define a closed subset of the compact space \(\mathbb T\times[1,2]\). Its projection to \(\mathbb T\) is compact and hence closed. Its inverse image under \(\pi\) is exactly \(R_\omega\). In particular, each indicator \({\bf1}_{R_\omega}(x)\) is Borel measurable in \(x\). Since \(\Omega\) is finite, it is also jointly measurable in \((\omega,x)\). For every stable center \(x\in G\), the inclusion \(B_\omega\subseteq B_\omega^+\) and Proposition~\ref{prop:raw-failure} give \[ \mathbb P(x\in R_\omega)\le2p. \] At the other centers the probability is at most \(1\). Interchanging a finite sum and an integral, and using Lemma~\ref{lem:stable}, we obtain \begin{align} \mathbb E\rho(R_\omega) &=\int_0^1\mathbb P(x\in R_\omega)\,dx \notag\\ &\le 2p\,m(G\cap[0,1])+m(G^c\cap[0,1]) \le2p+\rho(G^c)\le3p. \label{eq:exceptional-mean} \end{align} Thus \eqref{eq:open-mean} and \eqref{eq:exceptional-mean} imply \[ \mathbb E\bigl(\rho(B_\omega^+)+\rho(R_\omega)\bigr)\le5p. \] Choose one outcome whose summed density is at most this bound, and write its sets as \(B^+\) and \(R\). We have \begin{equation} \rho(B^+)+\rho(R)\le5p. \label{eq:chosen-outcome} \end{equation} It remains to cover every center in \(R\). Its image \(R_{\mathbb T}=\pi(R)\) is closed in the circle. If it is empty, put \(V=\varnothing\). Otherwise, for \(\varepsilon>0\), its open metric neighborhoods \[ V_\varepsilon^{\mathbb T} =\{z\in\mathbb T: \operatorname{dist}_{\mathbb T}(z,R_{\mathbb T}) <\varepsilon\} \] decrease to \(R_{\mathbb T}\) as \(\varepsilon\downarrow0\). Continuity of finite measure from above lets us choose \(\varepsilon>0\) so that the open periodic lift \(V=\pi^{-1}(V_\varepsilon^{\mathbb T})\) satisfies \[ R\subseteq V,\qquad \rho(V)\le\rho(R)+p. \] Set \(H=B^+\cup V\). It is open and periodic, and \eqref{eq:chosen-outcome} gives \[ \rho(H)\le\rho(B^+)+\rho(V) \le\rho(B^+)+\rho(R)+p\le6p. \] If \(x\notin R\), negating \eqref{eq:exceptional-centers} shows that for every \(t\in[1,2]\) some \(n\in\mathcal N\) satisfies \(x+t2^{-n}\in B^+\subseteq H\). If \(x\in R\), choose an integer \(n_0\ge1\) with \(2^{1-n_0}<\varepsilon\). For every \(t\in[1,2]\) and every \(n\ge n_0\), the circle distance satisfies \[ \operatorname{dist}_{\mathbb T} \bigl(\pi(x+t2^{-n}),R_{\mathbb T}\bigr) \le \operatorname{dist}_{\mathbb T} \bigl(\pi(x+t2^{-n}),\pi x\bigr) \le t2^{-n}<\varepsilon. \] Hence \(x+t2^{-n}\in V\subseteq H\). These indices are taken from the full dyadic tail. The two cases establish the required hit for every real \(x\) and every \(t\in[1,2]\), completing the lemma. \end{proof} ==== END The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/repair.tex ==== ==== BEGIN The-dyadic-case-of-the-Erdos-similarity-conjecture//sections/routing.tex ==== \section{Random routing and independent tests} \label{sec:routing} We now use the windows from the preceding section to construct a random periodic set \(B\) of expected density \(p\). Selectors send each spatial point to one leaf of the tree, where an independent terminal entry decides membership in \(B\). For a stable center, the first default choice on its path will provide several child windows in which to test nearby points. \subsection{The tables and the selected set} For each edge \(e=(P,i)\) with \(i\log(100M/p)/7$ works simultaneously for all larger $r$. The uniformity will allow us to enlarge windows as needed. We assign a window length $r_h$ to edges whose parent has height $h$. At the same time we compute the span $\sigma_h$ of all windows in a subtree of height $h$, including the gaps between them. Starting with $\sigma_0=0$, define successively for $h=1,\ldots,d$ \begin{equation} \begin{split} r_h&=\max\{r_*,g+\sigma_{h-1}\},\\ \sigma_h&= \begin{cases} Mr_1+(M-1)g,&h=1,\\ M(r_h+g+\sigma_{h-1})+(M-1)g,&h\ge2. \end{cases} \end{split} \label{eq:window-recursion} \end{equation} This is a finite bottom-up recursion: $r_h$ depends only on lengths already chosen below height $h$. In particular, the choice of $g$ did not depend on the sizes of these lengths. Now place the actual integer windows in the edge preorder, starting at $3$. An edge $e$ from height $h$ receives $W_e=\{a_e,\ldots,b_e\}$ with $b_e=a_e+r_h-1$; if $f$ immediately precedes $e$, set $a_e=b_f+g+1$. Put \[ \mathcal N=\bigcup_{e\in\mathcal T}W_e, \] where the union is over the edges of the tree. The span interpretation of \eqref{eq:window-recursion} follows by induction. At height $1$ there are $M$ windows and $M-1$ gaps. At height $h\ge2$, each child block consists of an incoming window of length $r_h$, one gap, and a subtree block of span $\sigma_{h-1}$; there are another $M-1$ gaps between these child blocks. For an edge $e=(P,i)$, let $b_e^*$ be the largest window endpoint among $e$ and all edges below $P_i$. If $P$ has height $h=1$, this block consists only of $e$, and its span is $r_1$. If $h\ge2$, its span is $r_h+g+\sigma_{h-1}$. Therefore in both cases \begin{equation} b_e^*-a_e+1\le r_h+g+\sigma_{h-1}\le2r_h. \label{eq:padding} \end{equation} In particular, \eqref{eq:entropy-budget} applies to every length $r_h$, while \eqref{eq:padding} bounds a whole child block in terms of its incoming window. All edges, including those with child index $M$, receive windows under this construction. The padding bound has a specific purpose: the range of window indices from $a_e$ through $b_e^*$ in a child block is controlled by the length of its incoming window. The gaps have a different role, established next: for most centers, translations indexed by a later window preserve the earlier cells used in the routing decisions. These are the two deterministic inputs to the routing and scale-counting arguments. \input{figures/windows} \subsection{Earlier grid cells remain unchanged} For every nonnegative integer $b$, define the periodic cell key \begin{equation} J_b(z)=\left\lfloor2^{b+2}\{z\}\right\rfloor, \qquad \{z\}=z-\lfloor z\rfloor. \label{eq:grid-key} \end{equation} Its boundaries on the real line form the lattice $\Gamma_b=2^{-(b+2)}\mathbb Z$. In particular, every integer is a boundary. Cells are half-open on the right, so a boundary belongs to the cell starting there. The grids are nested: for $B\ge b$, \begin{equation} J_b(z)=\left\lfloor\frac{J_B(z)}{2^{B-b}}\right\rfloor. \label{eq:grid-nesting} \end{equation} This follows by writing $2^{B+2}\{z\}=J_B(z)+\varepsilon$ with $0\le\varepsilon<1$ and dividing by the integer $2^{B-b}$. We seek centers for which translations indexed by one window preserve every earlier grid key. For a noninitial window $W_e$, let $b'$ be the endpoint of the immediately preceding window, and put $\delta_e=2^{1-a_e}$. Call $x$ \emph{stable} if, for every such window, \begin{equation} (x,x+\delta_e]\cap\Gamma_{b'}=\varnothing. \label{eq:stable-definition} \end{equation} Let $G$ denote the set of stable centers. The intervals in \eqref{eq:stable-definition} are taken on the real line, so crossings of an integer are included without a separate convention on the circle. \begin{lemma}\label{lem:stable} The set $G$ is measurable and $1$-periodic, and $\rho(G^c)