Geometric sequences and the Erdős similarity problem

Source and verification

This article explains family 084 of the October 6, 2026 OpenAI collection. The research conclusions are attributed to its manuscripts. The source review covers statements, introductions, and the proof mechanisms described below; it is not an independent verification of the complete proofs.

Overview

Positive-measure sets contain arbitrarily small copies of every finite pattern. The corresponding assertion for an infinite pattern is much more delicate. Family 084 claims that every fixed geometric sequence can be avoided by a compact subset of $[0,1]$ of measure arbitrarily close to one.

🏷️ Preliminaries: affine copies and universality

For a set $A\subset \mathbb{R}$, a nontrivial affine copy is $x+sA$ with $x\in \mathbb{R}$ and $s\ne 0$. Negative $s$ permits reflection. The set $A$ is measure universal if every measurable set of positive Lebesgue measure contains such a copy.

Every finite pattern is measure universal. To see this, take a subset $E$ of finite positive measure and a finite set $F$. Translation continuity in $L^1$ implies

$$\begin{array}{c}m(E\setminus (E-sa))⟶0\\ (s\to 0)\end{array}$$

for each $a\in F$. The union bound gives

$$\begin{array}{rl}& m\left(\bigcap _{a\in F}(E-sa)\right)\\ & \ge m(E)-\sum _{a\in F}m(E\setminus (E-sa))\\ & >0\end{array}$$

for sufficiently small nonzero $s$. Any point in the intersection gives the required copy. The finite sum is precisely where this proof stops working for an infinite pattern.

🏷️ The geometric-sequence claim

Fix $q\in (0,1)$ and write $G_q=\{q^n:n\ge 1\}$.

Family 084, as stated in the manuscripts

For every fixed $q\in (0,1)$ and $\eta \in (0,1)$, there is a compact $E_{q,\eta }\subset [0,1]$ with $m(E_{q,\eta })>1-\eta$ such that

$$\begin{array}{c}\forall x\in \mathbb{R}\forall s\ne 0\exists n\ge 1:\\ x+s{q}^n\notin E_{q,\eta }.\end{array}$$

The earlier companion treats $q=1/2$; the later manuscript treats every fixed ratio. (OpenAI, 2026) (OpenAI, 2026a)

The avoiding set may depend on $q$. The assertion is not one compact set simultaneously avoiding every ratio, and it does not address every infinite subset of the line. It controls all translations and all nonzero signed dilations of the chosen sequence.

🏷️ Small hitting sets instead of large avoiding sets

The construction works with an open set $U$ that meets every relevant affine copy, then takes $E=[0,1]\setminus U$. The target is to make $m(U\cap [0,1])$ small. Openness makes the final avoiding set compact.

The source first normalizes positive dilation factors to $t\in [1,2]$ and constructs a small-density periodic hitting set for that interval. A countable collection of dilations and reflections then treats all nonzero scales. The individual measure costs must be chosen summably so their union remains small.

There are two quantifier difficulties. A random construction can make a fixed copy unlikely to be missed, but there are uncountably many centers and scales. Moreover, a fine grid for the whole construction can introduce more candidate tests than the probability estimate can handle.

🏷️ The finite routing construction

The source assigns random binary tables to a finite ordered tree. At each node, the table values at a point’s grid cell determine its route to a child; a designated last child is selected by default when earlier tests fail. A separate sparse table at the terminal leaf determines membership in the random hitting set. Keeping that final decision separate preserves a small expected density.

Each edge receives a block of consecutive sequence indices. Gaps between blocks make later small translations unlikely to change earlier routing decisions. Within a block, the translated sequence points occupy distinct cells of the relevant fresh table, producing multiple independent chances of a hit.

The source’s key size condition makes an edge’s block long enough to dominate the following gap and its child subtree. Thus the number of scale representatives relevant to that block depends on its own length, rather than the finest grid anywhere in the tree. The branching parameter is then chosen so that the independent-test failure probability beats this local scale count.

For a general fixed $q$, the proof keeps nested dyadic grids but chooses their resolutions in relation to $q^{-b}/(1-q)$ at index $b$. The factor $1-q$ reflects the separation of consecutive geometric points. Constants and branching choices may therefore depend on $q$.

🏷️ Repairing every exceptional center

The probabilistic stage can leave a small set of exceptional centers. The final repair covers these centers by a small open neighborhood. Since $x+t{q}^n\to x$, any open neighborhood of an exceptional center contains a sufficiently late point of every normalized copy centered there. This converts an estimate for almost all or most centers into an assertion for every center.

This final use of convergence is elementary; producing a sufficiently small exceptional set uniformly in scale is the hard finite construction described above.

🏷️ Key steps and proof outlines

The random set has small density before any hitting estimate

The finite construction separates routing tables from terminal membership tables. At a fixed point $z$, reveal every routing table needed to determine its leaf and terminal address. Conditional on those revelations, the terminal entry at that address is still a fresh Bernoulli variable with success probability $p$. Therefore $P(z\in B)=p$, and Fubini gives expected periodic density $E\rho (B)=p$.

This separation is essential. If the same random bit both routed a point and decided membership, conditioning on the selected route could bias its probability of belonging to $B$. The construction instead reserves sparse terminal bits solely for membership. The density estimate is exact and independent of the complexity of the routing tree. (OpenAI, 2026b)

Stability and separation supply fresh independent trials

For a center $x$ and normalized scale $t$, consider $y_n(t)=x+t{q}^n$. An edge of the tree receives a block of indices $n$. Later blocks correspond to much smaller displacements from $x$. Gaps between blocks ensure that these displacements do not change routing keys already revealed at earlier levels, except when $x$ lies close to a grid boundary.

The exceptional centers are explicitly covered by short intervals adjacent to grid boundaries. Their total measure is the number of boundaries times the maximum displacement. Choosing the gap large enough makes this cost small. Outside that exceptional set, the earlier route is fixed for every relevant test point.

Within one block, the points $q^n$ are separated by $(1-q)q^n$ at their local scale. The fresh table’s grid is chosen finer than this separation, so different tests use different entries. Tables in different child subtrees are independent as well. Conditional on the center’s exposed route, these facts produce genuinely fresh trials. Independence is thus verified at the level of table addresses, not inferred merely from distinct sequence indices.

Local scale counting avoids an impossible global union bound

For fixed tables, a test depends only on finitely many grid keys. As $t$ varies over $[1,2]$, those keys change only at finitely many crossing values. One can therefore choose a finite list of scale representatives on which all test outcomes are constant.

The crucial point is which grid determines that list. For an edge block, only its own subtree is relevant. The preorder window construction makes the total span of that subtree comparable to the edge’s index-block length. Hence the scale-list size is controlled by the local block, not by the finest grid in the entire tree. The number of fresh trials also grows with that local block length. Choosing the branching parameter appropriately lets their exponentially small failure probability dominate the local scale count.

This is the proof mechanism that upgrades a fixed-scale probabilistic statement to simultaneous control of every normalized scale. A union bound based on the global finest mesh would be far larger and would not be justified by the same trial estimate.

Exceptional centers are repaired using the accumulation point

After selecting a favorable realization, the construction enlarges its hitting set slightly to make it open and covers the remaining exceptional centers by an open set of small measure. If $x$ lies in that repair set, openness gives an interval around $x$ contained in it. Since $t{q}^n\to 0$, all sufficiently late $x+t{q}^n$ lie in that interval. Thus the repair handles every scale at each exceptional center simultaneously.

This use of the accumulation point is more efficient than repairing each missed affine copy separately. The latter would involve uncountably many objects and would provide no useful measure budget. The source first controls the set of bad centers and only then invokes convergence of the geometric sequence.

Summable dilations produce one compact avoiding set

Every positive dilation can be written as $2^{k}t$ with $k\in \mathbb{Z}$ and $t\in [1,2]$. Apply the periodic construction with densities $p_k$ decreasing geometrically in $|k|$, and take the union of its dilates $2^k{H}_k$ and their reflections. Periodicity bounds the measure of each dilate inside $[0,1]$, including boundary-period errors. The chosen decay makes the total cost less than $\eta$.

The complement in $[0,1]$ is compact because the hitting union is open. Normalization and reflection ensure that every nonzero signed dilation is hit. The dyadic companion follows the same logic with the simpler spacing $2^{-n}$; the general-ratio proof changes grid resolutions and constants to account for $1-q$. Neither argument requires one construction to work uniformly as $q\uparrow 1$. (OpenAI, 2026a)

See Also

• Family 073: distance sets shows how dimension can force a geometric configuration, while this family asks what near-full measure can still avoid.
• Family 075: Fourier convergence also relies on making exceptional sets summable across a carefully chosen sequence of scales.

🏷️ Manuscripts and verification scope

• The geometric case of the Erdős similarity conjecture — the pinned manuscript; its directory supplies the source and citation metadata.
• The dyadic case of the Erdős similarity conjecture — the pinned manuscript; its directory supplies the source and citation metadata.

The repository provides a formalization-scope page. Its scope and comparator list cover the dyadic ratio q = 1/2. They do not document the later claim for every fixed geometric ratio. These are statements about the repository’s documentation: no Lean build, axiom audit, or comparator-equivalence check was performed for this article.

OpenAI collection gives the reading sequence and source policy; coverage register records which families have articles and which remain to be written.

🐻  OpenAI 2026a. The dyadic case of the Erdős similarity conjecture.

🐻  OpenAI 2026b. The geometric case of the Erdős similarity conjecture.