Some additive combinatorics problems in matrix rings

Abstract

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.

Gauss sums over some matrix groups

“…Also, using our new bounds (see Remarks 2.3 and 2.6 below), we study some additive combinatorics problems in matrix setting and improve two results of [3]. (See Section 3.2.…”

“…From now on, following [3], we always assume n 2. Let M n (F q ) and Z n (F q ) be the set of n × n matrices and the set of n × n singular matrices, respectively, over the finite field F q .…”

“…Ferguson, Hoffman, Luca, Ostafe and Shparlinski [3] showed that if A and B are sufficiently large, then …”

“…(See Corollaries 2.2 and 2.5 below.) By using several results from algebraic geometry, in particular, Skorobogatov's estimates of character sums along algebraic varieties [23], Ferguson, Hoffman, Luca, Ostafe and Shparlinski [3] also provided another upper bounds of the sums (1.2) and (1.3) (in the case χ = 1) and used them to study some additive combinatorics problems in matrix rings. Their bounds have been applied by us to study some uniform distribution properties of some matrix groups.…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…Very recently, Alon et al [5,6], using graph-theoretic methods, studied sum-free sets of order m in finite Abelian groups, and also, sum-free subsets of the set [1, n]. Additive combinatorics problems in matrix rings is another active area of research [53,55,67,82,83,114,125,133,134,183,184,206,298].…”