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

摘要

The Hardy-Littlewood majorant problem has a positive answer only for even integer exponents ρ, while there are counterexamples for all p (∈)2N. Montgomery conjectured that even among the idempotent polynomials there must exist counterexamples, i.e. there exist a 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 ＋e~(2πix) ± e~(2πi(k＋2)x) three-term character sums, used for ρ = 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 (∈)2N. Continuing this work here we demonstrate that even k = 3,4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G'~2/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