The geometric case of the Erdős similarity conjecture
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
For every fixed q∈(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.
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MANUSCRIPT 01
The geometric case of the Erdős similarity conjecture
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed q∈(0,1) and every \eta∈(0,1), we construct a compact set E_{q,\eta}⊆[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…
Full source abstract
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed q∈(0,1) and every \eta∈(0,1), we construct a compact set E_{q,\eta}⊆[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.
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The dyadic case of the Erdős similarity conjecture
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.
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