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The geometric case of the Erdős similarity conjecture

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\ge1\}

, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

Manuscripts in this family · 2

The geometric case of the Erdős similarity conjecture

Oct 5, 2026·geometric-erdos-similarity.pdf

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed

q\in(0,1)

and every

\eta\in(0,1)

, we construct a compact set

E_{q,\eta}\subseteq[0,1]

of measure greater than

1-\eta

containing no nontrivial affine copy of

\{q^n:n\ge1\}

, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.

Full abstract

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The dyadic case of the Erdős similarity conjecture

Sep 25, 2026·paper.pdf

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 Erdős similarity conjecture.

Full abstract

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