Doorn 2025 smooth sums small spacings

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lemma_1: Van Doorn and Everts's counting lemma: the number of integers 2^x p^y in the window from (1 + epsilon)^j to (p - delta)(1 + epsilon)^j is less than log x_j log(p - delta)/(log 2 log p) plus a constant c_p, for all j >= 0.

theorem: Van Doorn and Everts's main theorem: for every odd integer p > 1 there is a constant C_p such that every positive integer is a sum of distinct numbers 2^x p^y whose largest term is below C_p times the smallest, with C_p = 6 admissible for p = 3, while no constant smaller than p is admissible.

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Wouter van Doorn and Anneroos R. F. Everts, Smooth sums with small spacings. arXiv:2511.04585v1 [math.NT] (6 November 2025). Labels and pages below are those of this v1 edition.

The paper answers a question going back to Erdős (1992) and to Erdős and Lewin (1996): whether some constant C > 2 allows every positive integer n to be written as a sum of distinct 3-smooth integers b_1 < ... < b_r with b_r < C b_1. Its unnumbered main Theorem (p. 2) treats the sequence A_p of integers 2^x p^y for an odd integer p > 1: there is a constant C_p such that every positive integer is a sum of distinct elements of A_p with b_1 < ... < b_r < C_p b_1. One may take C_p = (1/2) F(4p) in general, where F is a product of rounded-down iterated base-2 logarithms (each at least 1) defined on p. 2, and C_p = 2p (resp. 2(p + 1)) when p - 1 (resp. p + 1) is a power of two; no constant smaller than p can replace C_p. For p = 3 this gives C_3 = 6 (abstract and p. 2). The lower bound (pp. 2--3) is a counting argument: Lemma 1 (p. 3), which the paper draws from the discussion in Lecture 5 of Hardy's Ramanujan, bounds the number of elements of A_p in a window [x_j, (p - delta) x_j), and summing over windows shows that for 1 < C < p almost all positive integers are not sums with b_r < C b_1. The existence part (pp. 3--7) tweaks and generalizes the explicit representation procedure of Blecksmith, McCallum and Selfridge (Amer. Math. Monthly, 1998), using a set S of Lemma 2 (p. 4) and a coefficient bound, Lemma 3 (p. 6). Section 3 (p. 8) improves C_p for some p by using multisets and leaves open whether C_p < cp for an absolute constant c. The paper also reports Cambie's computer check that C = 32/9 works for all n <= 10^5 (p. 2).

Source: https://arxiv.org/abs/2511.04585. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2511.04585), every other right reserved.

Read status: claims checked for the Theorem (p. 2) and Lemma 1 (p. 3), read clause by clause on the print; the proof of the Theorem (pp. 2--7) read for its structure only. Nothing here is independently reviewed.

Bears on. #845: the problem asks, for a constant C, whether the sums of distinct numbers 2^k 3^l with b_t <= C b_1 have density 0. The Theorem (p. 2) at p = 3 shows that every positive integer is such a sum with b_t < 6 b_1, and that for 1 < C < 3 almost all integers are not such sums with b_t < C b_1; it decides nothing for 3 <= C < 6.

Results.

• Theorem (p. 2): For every odd integer p > 1 there is C_p with every positive integer a sum of distinct elements of A_p = {2^x p^y} with b_1 < ... < b_r < C_p b_1; C_p = (1/2) F(4p) in general, 2p when p - 1 is a power of two, 2(p + 1) when p + 1 is; no constant below p works. For p = 3, C_3 = 6.
• Lemma 1 (p. 3): With x_j = (1 + epsilon)^j and X_j the number of elements of A_p in [x_j, (p - delta) x_j), there is a constant c_p with X_j < log(x_j) log(p - delta)/(log 2 log p) + c_p for all j >= 0.

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