BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sedanur Albayrak (University of Calgary) DTSTART:20230926T200000Z DTEND:20230926T210000Z DTSTAMP:20260423T021141Z UID:NTC/26 DESCRIPTION:Title: Qua ntitative Estimates for the Size of an Intersection of Sparse Automatic Se ts\nby Sedanur Albayrak (University of Calgary) as part of Lethbridge number theory and combinatorics seminar\n\nLecture held in University of L ethbridge: M1060 (Markin Hall).\n\nAbstract\nIn 1979\, Erdős conjectured that for $k \\geq 9$\, $2^k$ is not the sum of distinct powers of $3$. Tha t is\, the set of powers of two (which is $2$-automatic) and the $3$-autom atic set consisting of numbers\nwhose ternary expansions omit $2$ has fini te intersection. In the theory of automata\, a theorem of Cobham (1969) sa ys that if $k$ and $\\ell$ are two multiplicatively independent natural nu mbers then a subset of the natural numbers that is both $k$- and $\\ell$-a utomatic is eventually periodic. A multidimensional extension was later gi ven by Semenov (1977). Motivated by Erdős' conjecture and in light of Cob ham’s theorem\, we give a quantitative version of the Cobham-Semenov the orem for sparse automatic sets\, showing that the intersection of a sparse $k$-automatic subset of $\\mathbb{N}^d$ and a sparse $\\ell$-automatic su bset of $\\mathbb{N}^d$ is finite. Moreover\, we give effectively computab le upper bounds on the size of the intersection in terms of data from the automata that accept these sets.\n LOCATION:https://researchseminars.org/talk/NTC/26/ END:VEVENT END:VCALENDAR