The local sum conjecture in two dimensions

Abstract

In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture which will be accessbile to a broad readership.1 In the literature, oscillation indices tend to be defined as negative numbers. We will consider their absolute values and define them as positive numbers.2 In [5] a couple of minor auxilary conditions were also imposed. 3 Strictly speaking, when the dimension is large, this is only true if d(φ) > 1; see [2].