Akshat Mudgal

An additive combinatorial approach to Hua's lemma

An important problem in analytic number theory is Waring's problem, which concerns the number of ways in which a given natural number can be represented as a sum of s k^th powers, for some fixed natural numbers s,k. Most modern approaches to this problem often end up analysing solutions to equations of the form x_1^k + ... + x_s^k = x_{s+1}^k + ... + x_{2s}^k, with the variables x_1, ..., x_{2s} lying in some interval {1,2, ..., N}. In this talk, we study a more general setting, and so, we consider additive equations of the shape f(x_1) + ... + f(x_s) = f(x_{s+1}) + ... + f(x_{2s}), where f is a convex function with suitable properties and x_1, ..., x_{2s} lie in a finite set of integers. We will survey some recent results on this type of problem as well as highlight the various connections between this question and some other well-known conjectures in additive combinatorics.