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 $E\subset {\Bbb T}$ and for any $\gamma<\gamma_p$, there is an idempotent trigonometrical polynomial $f$ satisfying $\int_E |f|^p > \gamma \int_{\Bbb T} |f|^p$. 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.


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pp. 1065-1108
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