Concentration of the integral norm of idempotents
  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. 
Aline Bonami and Szil\'ard Gy. R\'ev\'esz