Integral Concentration of idempotent trigonometric polynomials with gaps

We prove that for all p>1/2 there exists a constant $\gamma_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $\gamma<\gamma_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > \gamma \int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $\gamma_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $\gamma_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows striking differences with the case p=2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener. We find sharper results for $0<p\leq 1$ when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones.