On the Hardy–Littlewood Majorant Problem
نویسندگان
چکیده
Let Λ ⊆ {1, . . . , N}, and let {an}n∈Λ be a sequence with |an| ≤ 1 for all n. It is easy to see that ∥∥∥∥ ∑ n∈Λ ane(nθ) ∥∥∥∥ p ≤ ∥∥∥∥ ∑ n∈Λ e(nθ) ∥∥∥∥ p for every even integer p. We give an example which shows that this statement can fail rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.
منابع مشابه
Special Quadrature Error Estimates and Their Application in the Hardy-littlewood Majorant Problem
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 there exist counterexamples even among idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with some four-term idempotents. However, Mockenhaupt conjectured that even the classical 1 + e ± e...
متن کاملRoth’s Theorem in the Primes
We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled Hardy-Littlewood majorant property. We derive this from a rather more general result which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a r...
متن کاملHardy-Littlewood varieties and semisimple groups
We are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for ...
متن کاملA Majorant Problem
Let f(z) akzk a 0 be analytlc in the unlt disc. Any k=O o Inflnlte complex vector e (eo,et,e2 ) such that lekl 1, k 0,1,2 induces a function re(Z) akekZk whlch is still analytic k=O In the unit disc. In this paper we study the problem of maximizing the p-means: over all possible vectors e and for values of r close to 0 and for all p<2. k It is proved that a maxlmlzlng function Is f,{z} -laoi + ...
متن کامل