Erdős similarity for fixed-ratio geometric sequences
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For a fixed ratio q∈(0,1), can a compact set E⊂[0,1] of measure arbitrarily close to one avoid every affine copy of {qⁿ:n≥1}? The dilation may have either sign; the avoiding set may depend on q.
Why it matters and what counts as progress
Why it matters: Each fixed geometric ratio extends beyond the dyadic sequence while remaining weaker than the full infinite-set similarity conjecture.
What counts as progress: Extend dyadic avoidance to each fixed geometric ratio or prove a construction uniform over a specified family of ratios.
How the problem developed
- 1974 · Open problem [1]
Paul Erdős Posed the similarity question asking whether every infinite subset of the line can be avoided by some positive-measure set under nontrivial affine copies.
- 5 Oct 2026 · Reported claim [1]
OpenAI internal model The manuscript claims the geometric-progression case for every fixed ratio q in (0,1): a compact set of measure arbitrarily close to one contains no translated nontrivial dilation of {q^n}.
AI contribution
The geometric case of the Erdős similarity conjecture
For each fixed q∈(0,1) and η∈(0,1), there is a compact set E in [0,1] of measure greater than 1−η containing no translated copy of {qⁿ:n≥1} under any nonzero real dilation of either sign. E may depend on q.
Sources
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Publication dates and editorial changes.
OpenAI reports Erdős similarity for every fixed geometric ratio