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

Link

https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=65518

*Abstract*: For a non-degenerate integral quadratic form *F*(*x*_{1},…,x_{d})
in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix
any compact subspace Ω⊂ℝ^{d }of the affine quadric *F*(*x*_{1},…,x_{d})
= 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*^{
-1}*m*)^{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* = (*x*_{1},…,x_{d}) of *F*(*x*) = *N* such that *x*_{i }≡⋋_{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*(*x*_{1},…,*x*_{4}) in 4 variables we prove
the same result if *N* ≥ (*r*^{ -1}*m*)^{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.