We prove that for all p>1/2 there exists a constant γp > 0 such that, for any symmetric measurable set of positive measure E C T and for any γ>γp, there is an idempotent trigonometrical polynomial f satisfying ∫_E | f |^p > γ ∫_T | f |_p. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of γp>0 for p > 1 and conjectured that it does not exists for p = 1.

Furthermore, we prove that one can take γp = 1 when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when p ≠ 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 ≤ 1 when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones.