Speaker: Jonah Klein, University of South Carolina at Columbia
Title: “Bounding the j-th smallest modulus of a covering system with distinct moduli”
Abstract: A covering system is a finite set of arithmetic progressions with the property that each integer belongs to at least one of them. Ever since their introduction by Erdos in 1950, covering systems have attracted a steady amount of interest. Of particular interest has been the minimum modulus problem, posed by Erdos in 1950. Erdos asked if there is a uniform bound on the smallest modulus ˝ of a covering system with distinct moduli. In 2015, Hough showed that this is the case, and gave a bound of 10^16. In 2021, Balister, Bollobas, Morris, Sahasrabudhe, and Tiba modified Hough’s approach, developing a new method that they coined the distortion method. Using this method, they reduced Hough’s bound to 616000. The distortion method has seen a number of applications in recent years, one of which is the focal point of this talk. In joint work with Dimitris Koukoulopoulos and Simon Lemieux, we showed that the j-th smallest modulus of a covering system with distinct moduli is bounded by exp(c log2 (j)/ log log(j)) for some absolute constant c. In this talk, we will give an overview of the distortion method and its recent applications. We will then show how it was used, in conjunction with a theorem of Crittenden and Vanden Eynden, to obtain a bound on the j-th smallest modulus of a covering system with distinct moduli.

The Discrete Math Seminar (DMS) is intended for Kennesaw State faculty working in the various areas of algebra, number theory, and discrete mathematics to get together to discuss their current work or related questions. Seminars often involve advanced mathematical knowledge. However, the seminars are open to anyone interested in attending. This talk will take place in person and virtually.

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