  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.
