Kiss 2022 generalized sidon sets perfect powers

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corollary_1: For every k >= 2 and epsilon > 0 there is a B_2[g] set of positive k-th powers with A(x) >> x^(1/k - epsilon), where the multiplicity g is not specified by the statement.

theorem_2: If for some 2 <= h <= k the number of representations of n as a sum of h distinct positive k-th powers is below n^eta for every eta > 0 and all large n, then for every epsilon > 0 some set of positive k-th powers has bounded h-fold representation function and counting function >> x^(1/k - epsilon).

theorem_3: For every k >= 2 there is h_0(k) = O(8^k k^2) such that for every h >= h_0(k) and every epsilon > 0 some set of positive k-th powers has bounded h-fold representation function and counting function >> x^(1/h - epsilon).

theorem_p2: The paper's unnumbered upper bounds: a B_h[g] set of k-th powers has A(x) << x^(min(1/k, 1/h)), and a B_2[g] set of squares has A(x) << x^(1/2) / (log x)^(1/4).

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Sándor Z. Kiss, Csaba Sándor, Generalized Sidon sets of perfect powers. The Ramanujan Journal 59 (2022), no. 2, 351-363. arXiv:2006.02783, doi:10.1007/s11139-022-00622-z. The copy read for this card is arXiv:2006.02783v1 (4 June 2020); the arXiv record names arXiv's non-exclusive distribution license (arXiv:2006.02783), every other right reserved.

Kiss and Sándor ask how dense a B_h[g] set of k-th powers can be, first showing by a counting argument that no exponent above min(1/k, 1/h) is possible. Theorem 2 proves, for some 2 <= h <= k and conditionally on a Hypothesis-K-type bound R_{(Z^+)^k, h}(n) < n^eta for every eta > 0 and large n, that for every epsilon > 0 there is A contained in the k-th powers with R_{A,h}(n) bounded and A(x) >> x^{1/k - epsilon}; Theorem 3 gives an unconditional version for h >= h_0(k) = O(8^k k^2) with A(x) >> x^{1/h - epsilon}. The hypothesis is verified for h = 2 (using the classical n^{o(1)} bound for even k and a divisor argument for odd k), yielding Corollary 1: for every k >= 2 and epsilon > 0 there is a B_2[g] set A of k-th powers with A(x) >> x^{1/k - epsilon}. The method is the Erdős-Rényi probabilistic method: each k-th power is kept independently with a suitable probability, and the Erdős-Tetali disjointness lemma with the Erdős-Rado Delta-system lemma shows that the representation function is then bounded with probability 1 (Lemma 6, p. 5); no elements are deleted. For problem 158 this is a forward-citation source and a quantifier barrier: specialized to squares, Corollary 1 gives a B_2[g] subset of the squares with A(x) >> x^{1/2 - epsilon}, but g is allowed to depend on epsilon, so it abandons the fixed multiplicity g = 2 that problem 158 demands and is not a partial resolution.

Source: https://arxiv.org/abs/2006.02783.

Bears on. #158: the unnumbered bound on p. 2 with k = h = 2 and g = 2 gives A(x) << x^(1/2) / (log x)^(1/4) for every B_2[2] set of squares, representations counted with a <= b as the problem counts them, so no set of squares is a counterexample; Corollary 1 with k = 2 gives B_2[g] sets of squares with A(x) >> x^(1/2 - epsilon) only for an unspecified g, which may depend on epsilon, so it is neither a counterexample nor a partial resolution. The paper does not mention the problem.

Results. Labels and pages are those of arXiv:2006.02783v1 (pp. 1-12).

• Unnumbered bounds (p. 2): any B_h[g] set A of k-th powers satisfies A(x) << x^{min(1/k, 1/h)}, and any B_2[g] set of squares satisfies A(x) << x^(1/2) / (log x)^(1/4).
• Theorem 2 (p. 3): if, for some 2 <= h <= k and every eta > 0, R_{(Z^+)^k, h}(n) < n^eta for all n >= n_0(eta), then for every epsilon > 0 there is A subset of the k-th powers with R_{A,h}(n) bounded and A(x) >> x^{1/k - epsilon}.
• Corollary 1 (p. 3): for every k >= 2 and epsilon > 0 there exists a B_2[g] set A of k-th powers with A(x) >> x^{1/k - epsilon}; g is not specified and may depend on epsilon.
• Theorem 3 (p. 3): for every k >= 2 there is h_0(k) = O(8^k k^2) such that for h >= h_0(k) and every epsilon > 0 some A of k-th powers has R_{A,h}(n) bounded and A(x) >> x^{1/h - epsilon}.

Not given pages: Conjecture 1 (p. 2), that for every k >= 1, h >= 2 and epsilon > 0 some B_h[g] set of k-th powers has A(x) >> x^{min(1/k, 1/h) - epsilon}; Problem 1 (p. 3), which asks whether some A of squares has R_{A,3}(n) bounded and A(x) >> x^{1/3 - epsilon} for every epsilon > 0, left open; and Theorem 1 (p. 2), Vu's theorem quoted from the literature, that for fixed k >= 2 and m > m_0(k) there is a basis A of k-th powers of order m with A(x) << x^{1/m} log^{1/m} x.

No file of this source is held. The edition read carries arXiv's license; the Ramanujan Journal edition, which Crossref's record lists under CC BY 4.0 (api.crossref.org), was not read.