Integral concentration of idempotent trigonometric polynomials with gaps

Abstract

Abstract. We prove that for all p > 1/2 there exists a constant γ p > 0 such that, for any symmetric measurable set of positive measure E ⊂ T and for any γ < γ p , 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 γ p > 0 for p > 1 and conjectured that it does not exists for p = 1.Furthermore, we prove that one can take γ p = 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. (2000): Primary 42A05. Secondary 42A16, 42A61, 42A55, 42A82, 42B05. Mathematics Subject Classification