The L^1-norm of exponential sums in Z^d

Details The L^1-norm of exponential sums in Z^d The L^1-norm of exponential sums in Z^d

2011-10-10

arxiv.org/abs/1110.2014v3

Mathematics

Combinatorics

13 pages. Section 5 has been modified. Conjecture 5.1 has been changed and is now called a question. The logarithmic lower bou

Scientific paper

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L^1-norm of exponential sums of sets in the d-dimensional grid Z^d. We show that the L^1-norm of F_A is considerably larger than log|A| when A is a subset of Z^d with multidimensional structure. We furthermore prove similar lower bounds for sets in Z, which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno & Smith and Konyagin.

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