Activity: Talk or presentation › Invited talk › science-to-science
Description
We study the number of 3-term arithmetic progressions (3-APs) in finite sets A={a1<⋯<an}⊂R of real numbers. The maximum is achieved when A itself forms an n-term arithmetic progression, in which case the number of 3-APs is of order Θ(n2). If instead we consider convex sets, i.e. the consecutive differences di=ai+1−ai form a strictly increasing sequence, it seems natural to suspect that in this case the number of 3-APs is significantly smaller. Indeed, their number is bounded by O(n53). However, we will also present a construction having Ω(n32) many 3-APs. We will further discuss connections to Jarník theorem on integral points on convex curves. This is joint work with Thomas F. Bloom and Oliver Roche-Newton.