On Mockenhoupt's Conjecture in the Hardy-Littlewood Majorant Problem

TL;DR

This paper advances the understanding of the Hardy-Littlewood majorant problem by proving Mockenhaupt's conjecture for specific three-term character sums for a broader range of p, refining analytical techniques for these estimates.

Contribution

The authors prove Mockenhaupt's conjecture for three-term character sums for p in the range 0 < p < 6, extending previous results and refining analytical methods.

Findings

Mockenhaupt's conjecture holds for k=3,4 cases.

Refined fourth order quadrature formulae improve approximation accuracy.

Detailed error estimates enhance the understanding of derivative approximations.

Abstract

The Hardy-Littlewood majorant problem has a positive answer only for expo- nents p which are even integers, while there are counterexamples for all p =2 2N. Montgomery conjectured that even among the idempotent polynomials there must exist some counterex- amples, i.e. there exist some finite set of characters and some ? signs with which the signed character sum has larger pth norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, Mockenhaupt conjectured that even the classical 1 + e2?ix ? e2?i(k+2)x three- term character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0; 1; 2, i.e. in the range 0 < p < 6, p =2 2N. Continuing this work here we demonstrate that even the k = 3; 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadra- ture formulae, finite estimation of G02=G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation.