A lower bound for the <i>L</i> ¹ norm of exponential sums

Abstract

Let f(x) be a trigonometric polynomial with N (≥2) non-zero coefficients of absolute value not less than 1. In this paper it is proved that the L1 norm of f exceeds a fixed positive multiple of (logN/log log N)½. This result improves a previous one due to H. Davenport and P. J. Cohen (the same bound with exponent ¼).