{"paper":{"title":"Geometric Block Exponents and a Uniform Mixed-Alphabet Sumset Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Johannes Hosle, Paata Ivanisvili","submitted_at":"2026-06-24T03:33:28Z","abstract_excerpt":"We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\\log 4/\\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\\ge1$, the dimension-free mixed-alphabet sumset bound \\[\n  |A+B|\\ge (|A||B|)^{p_0},\n  \\qquad\n  A\\subset\\{0,1\\}^d,\\quad\n  B\\subset\\{0,1,\\ldots,m\\}^d. \\] For every $m\\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger expo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25350","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.25350/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}