==== 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}
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\usepackage{booktabs}
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\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\newcommand{\R}{\mathbb R}
\newcommand{\T}{\mathbb T}
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\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)