The geometric case of the Erdős similarity conjecture

1 · The essentials

Can a set occupy almost all of an interval and still exclude a particular infinite pattern? Here the pattern is a list of numbers obtained by repeatedly multiplying by a fixed number $q$ between zero and one. The main result constructs a set inside $[0, 1]$ whose total length is as close to one as desired, yet which never contains the whole pattern. This remains true however the pattern is shifted, enlarged, shrunk, or reversed. The construction works for every fixed choice of $q$ , but the set is allowed to change with that choice. Thus, occupying almost all the interval does not force a set to contain even one complete shifted and rescaled version of this pattern. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine keeping almost all of a one-metre strip while removing enough material to obstruct an infinite row of marks whose gaps keep shrinking by a fixed factor. No matter how you move, resize, or reverse that row, at least one mark must fall outside the retained material. The image is imperfect because the result concerns infinitely many exact points, not physical marks with thickness.

Why it matters

The result resolves the geometric-progression case of the Erdős similarity conjecture for every ratio between zero and one. It separates two properties that might seem closely linked: having nearly full length and containing a prescribed infinite pattern. The exclusion is uniform over all translations and all nonzero real dilations, including negative ones.

2 · Going further

Fix $q \in (0, 1)$ and $η \in (0, 1)$ . The main result constructs a compact set $E_{q, η} \subseteq [0, 1]$ with Lebesgue measure greater than $1 - η$ such that, for every $a \in R$ and every $b \in R ∖ {0}$ , the set ${a + b q^{n} : n \geq 1}$ is not contained in $E_{q, η}$ . Here $a$ translates the pattern, while $b$ rescales it and reverses its direction when negative.

The source identifies this as the geometric-progression case of the Erdős similarity conjecture. Its central construction must meet two requirements at once: retain almost all the measure of the unit interval and obstruct every affine copy of the chosen sequence. The supplied abstracts state that such compact sets are constructed, but do not describe the mechanism of construction or the steps of the argument.

The companion manuscript treats the dyadic sequence ${2^{- n} : n \geq 1}$ , corresponding to $q = 1/2$ . It states the same near-full-measure and affine-avoidance conclusions, again allowing either sign of nonzero dilation. The main manuscript covers every fixed ratio in $(0, 1)$ , so its statement includes the dyadic case, without asserting that one set works for multiple ratios simultaneously.

Useful notions

Geometric sequences
Translations and rescalings
Lebesgue measure as length
Compact subsets of the real line
Quantifiers: every fixed ratio versus all ratios simultaneously

What was known before / what is new

Before

The source places these manuscripts within the Erdős similarity conjecture but gives no account of earlier results or historical development. The companion manuscript states the dyadic case; the supplied material does not specify a chronology between the two manuscripts.

New

The main manuscript establishes the stated avoidance result for every fixed geometric ratio $q \in (0, 1)$ . The avoiding set is compact and can have measure arbitrarily close to one. Its exclusion covers every translation and every nonzero real dilation, rather than only positive dilations.

Limits and assumptions

The hypotheses require a fixed $q \in (0, 1)$ and a tolerance $η \in (0, 1)$ ; the constructed set may depend on both. Zero dilation is excluded, and the result prevents containment of the entire infinite sequence, not necessarily of finite portions. No simultaneous assertion for different ratios is made. The main result has a Lean formalization, but the supplied material does not detail its scope or establish separate formal verification of every companion claim. The source reports resolution of the geometric case, not of the full Erdős similarity conjecture, and gives no further account of what remains open.

Mini glossary

Geometric sequence: A sequence in which each successive term is obtained by multiplying the preceding term by a fixed ratio.
Affine copy: A pattern obtained by replacing every point $x$ with $a + b x$ , where $a$ is a translation and $b \neq = 0$ is a dilation.
Lebesgue measure: The notion of length used to quantify how much of the unit interval the constructed set occupies.
Compact set: For a subset of the real line, a set that is closed and bounded.
Dyadic sequence: The geometric sequence ${2^{- n} : n \geq 1}$ , whose fixed ratio is $1/2$ .

Keywords : Erdős similarity conjecture · Geometric progressions · Affine avoidance

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every fixed $q \in (0, 1)$ , constructs compact subsets of $[0, 1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of ${q^{n} : n \geq 1}$ , with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/084.md

Manuscripts in this family

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The geometric case of the Erdős similarity conjecture

05/10/2026

To be verified

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed $q \in (0, 1)$ and every $η \in (0, 1)$ , we construct a compact set $E_{q, η} \subseteq [0, 1]$ of measure greater than $1 - η$ containing no nontrivial affine copy of ${q^{n} : n \geq 1}$ , for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.

PDF GitHub folder

The dyadic case of the Erdős similarity conjecture

25/09/2026

To be verified

We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence ${2^{- n} : n \geq 1}$ . The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.

PDF GitHub folder