Concentration of the Integral Norm of Idempotents

Summary

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 L ¹ concentration, while the above-mentioned paper deals with L ^p 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 L ¹ 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 L ¹ norm. The same problem is considered on the group $\mathbb{Z}/q\mathbb{Z}$, 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.