Gabdullin 2024 trigonometric polynomials frequencies set cubes

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theorem_1_1: Gabdullin and Konyagin's L4-L2 inequality for trigonometric polynomials whose frequencies are cubes n^3 with n in a short interval [N, N + N^(2/3-eps)], with an absolute implied constant times eps^(-1/4).

theorem_1_2: Gabdullin and Konyagin's theorem that the cubes n^3 with N <= n <= N + (0.5N)^(1/2) form a Sidon set, together with a family of equal sums of two cubes inside intervals [N, N + CN^(1/2)] for arbitrarily large N, which shows that the length is sharp up to the constant.

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Gabdullin, M. R. and Konyagin, S. V., Trigonometric polynomials with frequencies in the set of cubes. Math. Notes 115 (2024), no. 3-4, 336--340. https://doi.org/10.1134/S0001434624030052. The copy read for this card is the arXiv preprint arXiv:2311.14937v2 (8 March 2024); the theorem labels and page references below are the preprint's.

Theorem 1.1 (p. 2) shows that for any eps > 0 and any trigonometric polynomial f with frequencies in {n^3 : N <= n <= N + N^{2/3-eps}}, one has ||f||_4 << eps^{-1/4} ||f||2 with absolute implied constant (the abstract, p. 1), a Lambda_4-type inequality for cubes in short intervals. Theorem 1.2 (p. 2) shows that {n^3 : N <= n <= N + (0.5N)^{1/2}} is a Sidon set, and that this range is sharp up to the constant (0.5)^{1/2}. The proof of Theorem 1.1 (Section 2, pp. 3--4) writes u^3 + v^3 = m with u, v in the short interval, notes that u + v is then a divisor of 4m lying just below (4m)^{1/3}, and bounds the number of such divisors by a divisors-in-short-intervals estimate (Theorem 2.2, p. 4, quoted from Cilleruelo and Cordoba); Lemma 2.1 (p. 3) converts this representation bound into the L4 bound. The sharpness in Theorem 1.2 comes from an infinite family of solutions of x^3 + y^3 = z^3 + t^3 within intervals [N, N + CN^{1/2}] for arbitrarily large N, built from a Pell-type equation that generalizes Ramanujan's 1^3 + 12^3 = 9^3 + 10^3 (Section 3, pp. 4--5). Unlike squares, cubes carry no obstruction of the form ||sum{n<=N} e(n^2 x)||_4 ~ N^{1/2}(log N)^{1/4}, and the authors call it reasonable to conjecture that the set of all cubes is a Lambda_4 set (p. 2). Remark 2.3 (p. 4) conjectures a bounded count of divisors of m in [m^alpha, m^alpha + m^beta] for all 0 < beta < alpha < 1 and says it would raise the exponent 2/3 - eps of Theorem 1.1 to 1 - eps.

Source: https://arxiv.org/abs/2311.14937. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2311.14937), every other right reserved.

Bears on. #1206: the problem asks whether {1, 2^3, ..., N^3} contains a Sidon set of size >> N, and whether some set A of positive density has {a^3 : a in A} Sidon. Theorem 1.2 (p. 2) gives, among the cubes of 1, ..., M, a Sidon block of about (M/2)^{1/2} consecutive cubes, of order M^{1/2} rather than >> M, and shows that for an absolute constant C > 0 and infinitely many N the cubes of the integers in [N, N + CN^{1/2}] are not Sidon; it says nothing about cubes that are not consecutive and answers neither question. Theorem 1.1 (p. 2) is a Lambda_4-type inequality, weaker than the Sidon property, and is background only.

Results. Labels and pages are those of arXiv:2311.14937v2, whose PDF pages are numbered as printed.

• Theorem 1.1 (p. 2; proof pp. 3--4): for any eps > 0 and any f with frequencies in {n^3 : N <= n <= N + N^{2/3-eps}}, ||f||_4 << eps^{-1/4} ||f||_2 with an absolute implied constant. The page also records Remark 2.3 (p. 4).
• Theorem 1.2 (p. 2; proof pp. 4--5): {n^3 : N <= n <= N + (0.5N)^{1/2}} is a Sidon set, sharp up to the constant (0.5)^{1/2} in the sense that, for some absolute C > 0 and arbitrarily large N, the cubes of the integers in [N, N + CN^{1/2}] are not Sidon.

Read status. Claims checked for Theorems 1.1 and 1.2 against the print; the proofs were read but not checked step by step. Nothing is independently reviewed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.