{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T07:18:26Z","timestamp":1764314306490,"version":"3.46.0"},"reference-count":15,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"name":"European Union Seventh Framework Program","award":["FP7\/2007-2013"],"award-info":[{"award-number":["FP7\/2007-2013"]}]},{"name":"ERC","award":["291111"],"award-info":[{"award-number":["291111"]}]},{"name":"Engineering and Physical Sciences Research Council: EPSRC Dept. Award","award":["1.22"],"award-info":[{"award-number":["1.22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:p>Let G be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius r around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset X of G such that the pairwise product set [Formula: see text] is covered by at most k translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547\u2013594, doi: https:\/\/doi.org\/10.1007\/s00493-008-2271-7 ; T. Tao, Metric entropy analogues of sum set theory (2014), https:\/\/terrytao.wordpress.com\/2014\/03\/19\/metric-entropy-analogues-of-sum-set-theory\/ ] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc.\u00a025(1) (2012) 189\u2013243, doi: https:\/\/doi.org\/10.1090\/S0894-0347-2011-00708-X ], it was shown for the discrete case that, at the asymptotic limit of X finite but large, the \u201capproximateness\u201d (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on X replacing finiteness. In particular, if G has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].<\/jats:p>","DOI":"10.1142\/s0219061324500223","type":"journal-article","created":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T06:30:08Z","timestamp":1717137008000},"source":"Crossref","is-referenced-by-count":1,"title":["On metric approximate subgroups"],"prefix":"10.1142","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2761-6513","authenticated-orcid":false,"given":"Ehud","family":"Hrushovski","sequence":"first","affiliation":[{"name":"University of Oxford, Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, England"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9188-5128","authenticated-orcid":false,"given":"Arturo","family":"Rodr\u00edguez Fanlo","sequence":"additional","affiliation":[{"name":"Hebrew University of Jerusalem, Einstein Institute of Mathematics, Jerusalem 9190401, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,7,12]]},"reference":[{"key":"S0219061324500223BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-012-0043-9"},{"key":"S0219061324500223BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-015-0326-7"},{"volume-title":"Fractal Geometry: Mathematical Foundations and Applications","year":"1990","author":"Falconer K.","key":"S0219061324500223BIB003"},{"key":"S0219061324500223BIB004","series-title":"Translations of Mathematical Monographs","volume-title":"Foundations of a Structural Theory of Set Addition","volume":"37","author":"Freiman G. 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