Peter Bradshaw

Energy bounds for k-fold sums in very convex sets

A set $A = \{a_1 < . . . < a_N \}$ is considered convex if the adjacent differences $a_{i+1} − a_i$ form a monotone sequence. This property can be iterated to define sets which are “more” convex. There is a maxim in additive number theory that convex sets are not additively structured, and the “more” convex a set, the less additive structure it exhibits. In this talk, we will establish a new energy bound which supports this intuition. Let $T_k(A)$ be the number of solutions to $a_1 + . . . + a_k = a_{k+1} + . . . + a_{2k}$, where $a_i ∈ A$ for $1 ≤ i ≤ 2k$. Rather than using the common incidence geometry or Fourier analysis techniques, our methods are purely elementary and we prove new upper bounds for Tk(A) where A is a highly convex set. This is joint work with Brandon Hanson and Misha Rudnev.