The group ring of ℚ/ℤ and an application of a divisor problem

Haynes, Alan; Homma, Kosuke

doi:10.1090/S0002-9939-08-09624-X

The group ring of $\mathbb {Q}/\mathbb {Z}$ and an application of a divisor problem
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by Alan K. Haynes and Kosuke Homma PDF

Proc. Amer. Math. Soc. 137 (2009), 1285-1293 Request permission

Abstract:

First we prove some elementary but useful identities in the group ring of $\mathbb {Q}/\mathbb {Z}$. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.

References

Cristian Cobeli, Kevin Ford, and Alexandru Zaharescu, The jumping champions of the Farey series, Acta Arith. 110 (2003), no. 3, 259–274. MR 2008011, DOI 10.4064/aa110-3-5
Paul Erdös, Some remarks on number theory, Riveon Lematematika 9 (1955), 45–48 (Hebrew, with English summary). MR 73619
P. Èrdeš, An asymptotic inequality in the theory of numbers, Vestnik Leningrad. Univ. 15 (1960), no. 13, 41–49 (Russian, with English summary). MR 0126424
K. Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2008), to appear.
K. Ford, Integers with a divisor in $(y,2y]$, proceedings of Anatomy of Integers (Montreal, March 2006), to appear.
Glyn Harman, Metric number theory, London Mathematical Society Monographs. New Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558
A. Haynes, A $p$-adic version of the Duffin-Schaeffer Conjecture, preprint.
Terence Tao and Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012, DOI 10.1017/CBO9780511755149