A note on the Duffin-Schaeffer conjecture with slow divergence

作者：Aistleitner Christoph^*

来源：Bulletin of the London Mathematical Society, 2014, 46(1): 164-168.

DOI：10.1112/blms/bdt085

For a non-negative function psi : N -> R, let W(psi) denote the set of real numbers x for which the inequality vertical bar nx - a vertical bar < psi(n) has infinitely many coprime solutions (a, n). The Duffin-Schaeffer conjecture, one of the most important unsolved problems in metric number theory, asserts that W(psi) has full measure provided Sigma(infinity)(n=1) psi(n)phi(n)n(-1) = infinity. Recently Beresnevich, Harman, Haynes and Velani proved that W(psi) has full measure provided psi satisfies an extra divergence condition. In the present note, we establish a slow divergence counterpart of their result: W(psi) has full measure, provided Sigma(infinity)(n=1) psi(n)phi(n)n(-1) = infinity holds and additionally there exists some c > 0 such that the sum of psi(n)pi(n)n(-1) for n = 2(2h) vertical bar 1, ... , 2(2h+1) is at most ch(-1), for all h >= 1.

• 出版日期2014-2