Akshat Mudgal

Sums of Linear Transformations in higher dimensions

Given a finite subset A of integers and co-prime natural numbers q,s, we consider the set q.A + s.A, that is, the sum of dilates of A. In recent years, finding suitable lower bounds for the cardinality of such sets in terms of |A|, q and s has seen considerable activity. In 2014, Balog and Shakan found sharp estimates for the same, that were tight in both the main term as well as the error term. Subsequently, they considered this problem in higher dimensional integer lattices. In this talk, we present a short survey of these results including our own improvement in the higher dimensional setting.