Bloom 2025 integers small digits multiple bases

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Thomas F. Bloom, Ernie Croot, Integers with small digits in multiple bases. arXiv:2509.02835 (2025). The copy read for this card is the arXiv version stamped "arXiv:2509.02835v1 [math.NT] 2 Sep 2025" (23 pages). The arXiv record (https://arxiv.org/abs/2509.02835, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1 shows that for any r >= 1 and integers g_1,...,g_r >= 2 with g_i^{a_i} != g_j^{b_j} for all i != j and all integers a_i, b_j >= 1, and weights kappa_1,...,kappa_r in (0,1] satisfying sum_j log_{g_j}(320 r^5 / kappa_j) < 1/(2r), for every epsilon > 0 there are infinitely many n such that for every j all but at most epsilon log n of the base-g_j digits of n are < kappa_j g_j; the condition holds whenever the bases are large in terms of r and the kappa_j. The proof in fact produces such an n in every interval [N, exp(O(epsilon^{-1-o(1)}))N] with N large enough in terms of epsilon, the g_j and the kappa_j, though Theorem 1 is ineffective and gives no bound on the smallest such n. The paper frames this as a weak form of Conjecture 1, a Pomerance-style heuristic prediction that the condition sum_j log_{g_j}(g_j / ceil(kappa_j g_j)) < 1 should suffice for all digits to be small, and it improves earlier work of Croot, Mousavi and Schmidt both quantitatively and qualitatively. By Kummer's criterion a prime p does not divide binomial(2n, n) exactly when every base-p digit of n is < p/2 (p. 1), so Graham's conjecture that infinitely many binomial(2n, n) are coprime to 105 = 3 * 5 * 7 is the case g = (3,5,7), kappa = 1/2 of Conjecture 1, where the heuristic sum is 0.974... < 1 (p. 2). From Theorem 1 the paper derives Corollary 1 (p. 3), which it calls a weak version of Graham's conjecture: for r >= 3 primes p_1, ..., p_r, all large enough in terms of r, and every epsilon > 0, infinitely many n have binomial(2n, n) = n_1 n_2 with n_1 prime to p_1 ... p_r and n_2 <= n^epsilon. This is the paper's bearing on problem 376: it does not settle the coprimality question, since Theorem 1 needs bases large enough for its condition, which fails for 3, 5 and 7 (for r = 3 and kappa_j = 1/2 the paper calculates that it holds once every base is at least 10^94, p. 3), and even for large bases only almost all digits (all but epsilon log n) are shown to be small; it establishes the corresponding multiple-base small-digit phenomenon for sufficiently large bases.

Source: https://arxiv.org/abs/2509.02835.

Bears on. #376

Results to transcribe.

• Conjecture 1: For bases g_1,...,g_r with no common power and weights kappa_j in (0,1] satisfying sum_j log_{g_j}(g_j / ceil(kappa_j g_j)) < 1, there should be infinitely many n with every base-g_j digit of n less than kappa_j g_j; Graham's conjecture is the case (3,5,7) with kappa = 1/2.
• Theorem 1: Under the stronger condition sum_j log_{g_j}(320 r^5 / kappa_j) < 1/(2r) (satisfied for all sufficiently large bases), for every epsilon > 0 there are infinitely many n such that for each j all but at most epsilon log n of the base-g_j digits of n are < kappa_j g_j; the proof finds such n in every interval [N, exp(O(epsilon^{-1-o(1)}))N] with N large in terms of epsilon, the g_j and the kappa_j, but the result is ineffective.