On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

Abstract

The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexamples for all p / ∈ 2N. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exist some finite set of characters and some ± signs with which the signed character sum has larger p th 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 threeterm 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 / ∈ 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 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. (2000): Primary 42A05. Mathematics Subject Classification

Special quadrature error estimates and their application in the hardy-littlewood majorant problem

“…His argument hinted that some numerical analysis may be used in the proof, but we could not complete the solution along those lines. Nevertheless, we have proved this conjecture for k = 0, 1, 2 in [7] and later even to k = 3, 4 in [8].…”

“…This has been recently proved by Mockenhaupt and Schlag in [10]. Their example is a four-term idempotent f and a signed version of it for g. For more details and explanations of methods and results see [7,8] and the references therein.…”

“…Although we do not detail the estimates of the computational error of applying spreadsheets and functions from Microsoft Excel tables, it is clear that under this step number size our calculations are reliable well within the error bounds. For a more detailed error analysis of that sort, which similarly applies here, too, see our previous work [7], in particular footnote 3 on page 141 and the discussion around formula (22), and see also the comments in the introduction of [8].…”

“…For larger k, however, in [7,8] we used function calculus and support our analysis by numerical integration and error estimates where necessary. Naturally, these methods are getting computationally more and more involved when k is getting larger.…”

“…We keep using the fourth order quadrature formula, presented and explained in [8], see [8,Lemma 5]. However, another new argument also has to be invoked for k = 5 compared to k = 3, 4, because in this case the analytic scheme of proving fixed signs of certain derivatives simply break down.…”