Concentration of the integral norm of idempotents

TL;DR

This paper investigates the $L^1$ concentration of idempotent trigonometric polynomials, achieving a high concentration constant that challenges previous conjectures and extends analysis to finite groups using recent additive number theory results.

Contribution

It provides new bounds for $L^1$ concentration of idempotents and applies advanced additive combinatorics techniques to analyze their behavior on groups.

Findings

Achieved a concentration constant $oldsymbol{964}$ in $L^1$ norm for idempotents.

Contradicted the conjecture that no positive $L^1$ concentration exists for idempotents.

Showed failure of absolute integral concentration on finite cyclic groups, supporting the conjecture's negation.

Abstract

This is a companion paper of a recent one, entitled {\sl 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 two-fold. At the first place we try to explain methods and results, and give further straightforward corollaries. On the other hand, we push forward the methods to obtain a better constant for the possible concentration (in $L^1$ norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level $\gamma_1>0.96$, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in $L^1$ 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.