Concentration of the integral norm of idempotents

Abstract

This chapter is a companion of a recent paper, entitled Integral concentration of idempotent trigonometric polynomials with gaps. New results of the present work concern L1 concentration, while the above-mentioned paper deals with Lp concentration. Our aim here is twofold. First we try to explain methods and results, and give further straightforward corollaries. On the other hand, we also push forward the methods to obtain a better constant for the possible concentration (in the L1 norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level γ1 > 0. 96, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in the L1 norm. The same problem is considered on the group Z/qZ, with q, say, a prime number. There, the property of absolute integral concentration of idempotent polynomials fails, which is in a way a positive answer to the conjecture mentioned above. Our proof uses recent results of B. Green and S. Konyagin on the Littlewood problem.