{"paper":{"title":"On asymptotic formulae in some sum-product questions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ilya D. Shkredov","submitted_at":"2018-02-25T19:23:08Z","abstract_excerpt":"In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\\mathbf{F}_p$, as well as the growth result in ${\\rm SL}_2 (\\mathbf{F}_p)$ due to Helfgott. Here some of our applications:\n  $\\bullet~$ a new bound for the number of the solutions to the equation $(a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $\\,a_i, a'_i\\in A$, $A$ is an arbitrary subset of $\\mathbf{F}_p$,\n  $\\bullet~$ a new effective bound for multilinear exponential sums of Bourgain,\n  $\\bullet~$ an asymptotic analogue"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09066","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"}