{"paper":{"title":"Resonance between Cantor sets","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CA","authors_text":"Pablo Shmerkin, Yuval Peres","submitted_at":"2007-05-18T04:08:27Z","abstract_excerpt":"Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\\log b/\\log a$ is irrational, then \\[ \\dim(C_a+C_b) = \\min(\\dim(C_a) + \\dim(C_b),1), \\] where $\\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\\dim(K)+\\dim(K') \\le 1$ (``geometric resonance''), then there exists $r<1$ such that all contrac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.2628","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}