Distances sets that are a shift of the integers and Fourier basis for planar convex sets

Alex Iosevich, Philippe Jaming

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta (A)$ satisfying $\Delta (A)\subset {\mathbb{Z}}^{+}+s$ for suitable $s$ than $A$ has at most 3 elements. This result…