Thomas Bloom: Sets with small l^1 Fourier norm

Abstract.

A famous conjecture of Littlewood states that the Fourier transform of every set of N integers has l^1 norm at least log(N), up to a constant multiplicative factor. This was proved independently by McGehee-Pigno-Smith and Konyagin in the 1980s. This lower bound is the best possible, as it is achieved by an arithmetic progression. An interesting question, especially from the perspective of additive combinatorics, is the 'inverse problem': what can we say about sets which are close to optimal, say with l^1 norm at most 100 log(N)? I will discuss an inverse result of this type, showing that any such set must correlate with a long arithmetic progression, and that such sets contain arbitrarily long arithmetic progressions.