BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Simon Machado (University of Cambridge) DTSTART:20220407T141500Z DTEND:20220407T160000Z DTSTAMP:20260422T065705Z UID:CJCS/61 DESCRIPTION:Title: Wh en are discrete subsets of Lie groups approximate subgroups ? Around a the orem of Lagarias.\nby Simon Machado (University of Cambridge) as part of Copenhagen-Jerusalem Combinatorics Seminar\n\n\nAbstract\nMeyer sets ar e fascinating objects: they are aperiodic subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic order. Sets of vertices of the P enrose tiling (P3) and Pisot—Vijarayaghavan numbers of a real number fie ld are some of the most well-known examples. In modern lingo\, they can be defined as the discrete and co-compact approximate subgroups of Euclidean spaces.\n\nLagarias found an elegant characterisation of Meyer sets: they are those subsets X of a Euclidean space E that are relatively dense (any point of E is within uniformly bounded distance of X) and whose Minkowski difference X - X is uniformly discrete (the distance between any two poin ts is uniformly bounded below). In essence\, his theorem provides a charac terisation of discrete approximate subgroups that is analogous to the Plun necke—Ruzsa theorem about sets of small doubling.\n\nI will discuss the general theory of Meyer sets and state Lagarias theorem. I will explain ho w additive combinatorics can be used to extend Lagarias result to discrete subsets of amenable groups. Going beyond the framework of amenable groups \, I will talk about how one can use simple ideas from additive combinator ics in combination with powerful tools from ergodic theory - such as Zimme r’s cocycle superrigidity - to generalise Lagarias theorem to discrete s ubsets of $SL_n(\\mathbb{R})$ for n > 2.\n LOCATION:https://researchseminars.org/talk/CJCS/61/ END:VEVENT END:VCALENDAR