HOW TO CONCENTRATE IDEMPOTENTS

Call a sum of exponentials of the form f(x) = exp (2πiN^sub 1^x) + exp (2πiN^sub 2^x) + . . . + exp (2πiN^sub m^x), where the N^sub k^ are distinct integers, an idempotent. We have L^sup p^ interval concentration if there is a positive constant a, depending only on p, such that for each interval I ⊂ [0, 1] there is an idempotent f so that ∫^sub I^ |f (x)|^sup p^ dx/ ∫^sup 1^^sub 0^ |f (x)|^sup p^ dx > a. We will explain how to produce such concentration for each p > 0. The origin of this question and the history of the development of its solution will be surveyed. [PUBLICATION ABSTRACT]