{"id":"openai-math/The-geometric-case-of-the-Erdos-similarity-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026","tags":["geometry","measure-theory","erdos-similarity","additive-combinatorics"],"score":0,"title":"The geometric case of the Erdős similarity conjecture","domain":"Real and complex analysis","family":"084","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"The geometric case of the Erdős similarity conjecture"}