Three-term idempotent counterexamples in the Hardy–Littlewood majorant problem

The Hardy–Littlewood majorant problem was raised in the 30ʼs and it can be formulated as the question whether ∫ | f | p ⩾ ∫ | g | p whenever f ˆ ⩾ | g ˆ | . It has a positive answer only for exponents p which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some ± signs with which the signed exponential sum has larger pth norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, a natural question is if even the classical 1 + e 2 π i x ± e 2 π i ( k + 2 ) x three-term exponential 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. We investigate the sharpened question and show that at least in certain cases there indeed exist three-term idempotent counterexamples in the Hardy–Littlewood majorant problem; that is we have for 0 < p < 6 , p ∉ 2 N , ∫ 0 1 2 | 1 + e 2 π i x − e 2 π i ( [ p 2 ] + 2 ) x | p > ∫ 0 1 2 | 1 + e 2 π i x + e 2 π i ( [ p 2 ] + 2 ) x | p . The proof combines delicate calculus with numerical integration and precise error estimates.