Barak Weiss: New bounds on lattice covering volumes, and nearly uniform covers (NTWS 173)
Channel: Number Theory Web Seminar
Date: 2023-05-28

Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called “Geometry of Numbers”, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.


Link to slides: https://drive.google.com/file/d/14-1paPSowG0QZ55BeKvgDao4fVLRgKCC

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, May 25, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 26, 2023 (1am AEST, 3am NZST)