TY - JOUR
T1 - On the metric theory of approximations by reduced fractions: A quantitative Koukoulopoulos-Maynard theorem
AU - Aistleitner, Christoph
AU - Borda, Bence
AU - Hauke, Manuel
N1 - Funding Information:
CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945, I-5554, P-34763, P-35322 and Y-901. BB is supported by the Austrian Science Fund (FWF), project F-5510. We wish to thank the referee for a very careful reading of our paper and for many helpful comments.
Publisher Copyright:
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.
PY - 2023/2/3
Y1 - 2023/2/3
N2 - Let ψ:N→[0,1/2] be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals α there are infinitely many coprime solutions (p,q) to the inequality |α - p/q|<ψ(q)/q, provided that the series Σq=1Q φ(q)ψ(q)/q is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all α the number of coprime solutions (p,q), subject to q≤Q, is of asymptotic order Σq=1Q 2φ(q)ψ(q)/q. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic input on the 'anatomy of integers'. The key phenomenon is that the system of approximation sets exhibits 'asymptotic independence on average' as the total mass of the set system increases.
AB - Let ψ:N→[0,1/2] be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals α there are infinitely many coprime solutions (p,q) to the inequality |α - p/q|<ψ(q)/q, provided that the series Σq=1Q φ(q)ψ(q)/q is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all α the number of coprime solutions (p,q), subject to q≤Q, is of asymptotic order Σq=1Q 2φ(q)ψ(q)/q. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic input on the 'anatomy of integers'. The key phenomenon is that the system of approximation sets exhibits 'asymptotic independence on average' as the total mass of the set system increases.
KW - Diophantine approximation
KW - Duffin-Schaeffer conjecture
KW - Koukoulopoulos- Maynard theorem
KW - metric number theory
UR - http://www.scopus.com/inward/record.url?scp=85150734831&partnerID=8YFLogxK
U2 - 10.1112/S0010437X22007837
DO - 10.1112/S0010437X22007837
M3 - Article
AN - SCOPUS:85150734831
SN - 0010-437X
VL - 159
SP - 207
EP - 231
JO - Compositio Mathematica
JF - Compositio Mathematica
IS - 2
ER -