# DnA Seminar, Naser Talebizadeh (Princeton): Optimal strong approximation for quadratic forms

Tuesday, November 10, 2015
12:00 PM - 1:00 PM (ET)
Exley Science Center (Tower)
Event Type
Seminar/Colloquium
Contact
Caryn Canalia
Department
Mathematics and Computer Science
https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=65518

Abstract:  For a non-degenerate integral quadratic form F(x1,…,xd) in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace Ω⊂ℝd of the affine quadric F(x1,…,xd) = 1. Suppose that we are given a small ball B of radius 0 < r < 1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies N » (r -1m)4+ for any

> 0. Finally assume that we are given an integral vector (1,…,d) mod m.  Then we show that there exists an integral solution x = (x1,…,xd) of F(x) = N such that xi i mod m and   B, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form F(x1,…,x4) in 4 variables we prove the same result if N ≥ (r -1m)6+ and some non-singular local conditions for N are satisfied. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form F(X) in 4 variables with the optimal exponent 4.