Garaev 2026 sidon sets squares cubes quartics short

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theorem_1: States that for every positive integer N the squares of the integers n with N at most n and n less than N + (8N+8)^(1/2) + 2 form a Sidon set, and that for infinitely many N the same set with n allowed to equal that endpoint is not a Sidon set.

theorem_2: States that for each fixed eps > 0 and every N larger than some N_0(eps), the squares of the integers from N to N + ((8+eps)N)^(1/2) do not form a Sidon set, so the constant 8 of Theorem 1 cannot be enlarged for large N.

theorem_3: States that for every positive integer N the cubes of the integers n with N at most n and n less than N + (38N/3 + 1297/36)^(1/2) + 19/6 form a Sidon set, and that for infinitely many N the same set with n allowed to equal that endpoint is not a Sidon set.

theorem_4: States that there is an absolute constant c > 0 such that for every positive integer N the cubes of the integers from N to N + cN^(2/3) do not form a Sidon set, so x^3 + y^3 = z^3 + t^3 always has a non-trivial solution in that range.

theorem_5: States that for every eps > 0 there are infinitely many positive integers N for which the cubes of the integers from N to N + N^(4/7-eps) form a Sidon set, so for cubes the sharp endpoint of Theorem 3 is not the right length for every N.

theorem_6: States that there is an absolute constant c > 0 such that for every positive integer N the fourth powers of the integers from N to N + cN^(3/5) form a Sidon set, improving the exponent 1/2 that the paper calls not difficult to obtain.

theorem_7: States that there is an absolute constant c > 0 such that for every positive integer N the fourth powers of the integers from N to N + cN^(12/13) do not form a Sidon set, so x^4 + y^4 = z^4 + t^4 always has a non-trivial solution in that range.

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M. Z. Garaev, F. M. Garayev, S. V. Konyagin, On Sidon sets with squares, cubes and quartics in short intervals. arXiv:2602.08807 (2026). The copy read for this card is arXiv:2602.08807v2 (6 May 2026).

The paper refines results of Gabdullin and of Gabdullin-Konyagin on how long an interval can be while the squares, cubes, or fourth powers of the integers in it still form a Sidon set. For squares, Theorem 1 (pp. 2-3) shows that {n^2 : N <= n < N + (8N+8)^{1/2} + 2} is a Sidon set for every positive integer N, and that the strict inequality cannot be replaced by <=; Corollary 1 (p. 3) gives the endpoint (8N)^{1/2} + 2 with both constants sharp, and Theorem 2 (p. 3) shows that for each eps > 0 the set {n^2 : N <= n <= N + ((8+eps)N)^{1/2}} is not a Sidon set once N > N_0(eps). For cubes, Theorem 3 (p. 3) shows that {n^3 : N <= n < N + (38N/3 + 1297/36)^{1/2} + 19/6} is a Sidon set for every positive integer N, and that for infinitely many N the same set with <= in place of < is not; Corollary 2 (p. 3) gives the endpoint (38N/3)^{1/2} + 19/6 with both constants sharp. Theorem 4 (p. 3) shows that {n^3 : N <= n <= N + cN^{2/3}} is never a Sidon set, for an absolute constant c > 0, while Theorem 5 (p. 4) shows that for every eps > 0 there are infinitely many N for which {n^3 : N <= n <= N + N^{4/7-eps}} is a Sidon set, so unlike squares the cube length is not of order N^{1/2} for every N. For fourth powers, Theorem 6 (p. 4) shows that {n^4 : N <= n <= N + cN^{3/5}} is a Sidon set for every N, and Theorem 7 (p. 4) that {n^4 : N <= n <= N + cN^{12/13}} never is, each for an absolute constant c > 0. The results sharpen Gabdullin's statement that {n^2 : N <= n <= N + (8N)^{1/2}} is a Sidon set and Gabdullin and Konyagin's that {n^3 : N <= n <= N + (0.5N)^{1/2}} is one (both reported on p. 2). The cube results come from generalized Pell equations (Sections 5 and 7) and from the Euler-Binet parametrization (Section 6); the quartic bound of Theorem 7 comes from Euler's parametric solution (Section 9).

Source: https://arxiv.org/abs/2602.08807. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2602.08807), every other right reserved.

Bears on.

• Problem 1206: background only. The problem asks whether {1, 2^3, ..., N^3} contains a Sidon set of size

  N. Theorems 3 and 5 and Corollary 2 give Sidon sets of consecutive cubes, and Theorem 4 limits them: a consequence noted on the Theorem 4 page, not stated in the paper, is that blocks of consecutive cubes inside {1, 2^3, ..., N^3} that are Sidon sets have O(N^{2/3}) elements. The paper treats only such blocks and does not address general Sidon subsets of the cubes.

• Problem 773: background only. The problem asks for the largest Sidon subset of {1, 2^2, ..., N^2}. Theorems 1 and 2 determine, to within o(N^{1/2}), the length of the longest Sidon block of consecutive squares starting at a given square; they say nothing about general subsets.

Result pages.

• Theorem 1 (pp. 2-3): the squares n^2 with N <= n < N + (8N+8)^{1/2} + 2 form a Sidon set for every N, and the endpoint cannot be included; the page also records Corollary 1 (p. 3).
• Theorem 2 (p. 3): for fixed eps > 0 and N > N_0(eps), the squares n^2 with N <= n <= N + ((8+eps)N)^{1/2} are not a Sidon set.
• Theorem 3 (p. 3): the cubes n^3 with N <= n < N + (38N/3 + 1297/36)^{1/2} + 19/6 form a Sidon set for every N, and the endpoint cannot be included; the page also records Corollary 2 (p. 3).
• Theorem 4 (p. 3): for an absolute constant c > 0, the cubes n^3 with N <= n <= N + cN^{2/3} are never a Sidon set.
• Theorem 5 (p. 4): for every eps > 0 and infinitely many N, the cubes n^3 with N <= n <= N + N^{4/7-eps} form a Sidon set.
• Theorem 6 (p. 4): for an absolute constant c > 0, the fourth powers n^4 with N <= n <= N + cN^{3/5} form a Sidon set for every N.
• Theorem 7 (p. 4): for an absolute constant c > 0, the fourth powers n^4 with N <= n <= N + cN^{12/13} are never a Sidon set.

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