Abstract:We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Teräväinen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp.
From: Cheuk Fung Lau [view email]
[v1]
Thu, 16 Apr 2026 14:08:26 UTC (68 KB)
[v2]
Wed, 24 Jun 2026 16:50:52 UTC (71 KB)
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