A note on the Duffin-Schaeffer conjecture with slow divergence

Aistleitner, Christoph

doi:10.1112/blms/bdt085

Cited by 15 publications

(54 citation statements)

References 10 publications

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“…We should remark that the above kind of arguments was first observed by Haynes, Pollington and Velani ( [22], see also [1,4]). Hence to study the Duffin-Schaeffer conjecture it brings no harm for us to assume h S h = ∞ together with:…”

Section: Upper Boundssupporting

confidence: 60%

“…Several recent progresses on the classical Duffin-Schaeffer conjecture ( [1,4,25]) can be naturally transferred into new results on the p-adic version of the DuffinSchaeffer conjecture via a lemma of Haynes ([21, Lemma 3]), but we will not pursue this direction in the paper.…”

Section: Introduction To the Duffin-schaeffer Conjecturementioning

confidence: 99%

“…According to the first Borel-Cantelli lemma, we see that if the series Although various partial results are known (see [19] for details and references before 2000 and [1,4,22,25] for recent progresses), the full conjecture represents one of the most fundamental unsolved problems in metric number theory.…”

Section: Introduction To the Duffin-schaeffer Conjecturementioning

confidence: 99%

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The Duffin–schaeffer‐type Conjectures in Various Local Fields

2016

Mathematika

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Abstract. This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that λ(∩ ∞ m=1 ∪ ∞ n=m E n ) = 1 if and only if n λ(E n ) = ∞, where λ denotes the Lebesgue measure on R/Z,ψ is any non-negative arithmetical function. Instead of studying ∩ ∞ m=1 ∪∞ n=m E n we introduce an even fundamental object ∪ ∞ n=1 E n and conjecture there exists a universal constant C > 0 such thatIt is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena are found in the fields of p-adic numbers and formal Laurent series. As a byproduct, we answer conditionally a question of Haynes by showing that one can always use the quasi-independence on average method to deduce λ(∩ ∞ m=1 ∪ ∞ n=m E n ) = 1 as long as the Duffin-Schaeffer conjecture is true. We also show among several others that two conjectures of Haynes, Pollington and Velani are equivalent to the Duffin-Schaeffer conjecture, and introduce for the first time a weighted version of the second Borel-Cantelli lemma to the study of the Duffin-Schaeffer conjecture. Introduction to the Duffin-Schaeffer conjectureThroughout the paper we use the following notations:• p denotes a prime number, • n, h denote positive integers, • ϕ(n) denotes the Euler phi function, • λ denotes the Lebesgue measure on R/Z, r) (B(x, r)) denotes the open (closed) ball with center x and radius r in a given metric space.2000 Mathematics Subject Classification. 11J83.

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Section: Upper Boundssupporting

confidence: 60%

Section: Introduction To the Duffin-schaeffer Conjecturementioning

confidence: 99%

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The Duffin–schaeffer‐type Conjectures in Various Local Fields

2016

Mathematika

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“…Remark. Theorem 1.8 (and the two results that follow it) may be considered in the context of recent works [1,2,4,13], which establish the Duffin-Scheffer conjecture under "extra divergence" assumptions. In the most recent of these, Aistleitner et al show that if for some ε > 0 the sum…”

mentioning

confidence: 98%

“…The Duffin-Schaeffer conjecture remains open, 1 and it is one of the most pursued problems in metric number theory. The difficulty in it arises from the fact that the sets W ( )-and also W ( )-are lim sups of sequences of sets that are not in general independent.…”

mentioning

confidence: 99%

Khintchine's Theorem With Random Fractions

Ramírez

2019

Mathematika

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We prove versions of Khintchine's theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin–Schaeffer conjecture (1941) and are likely to be more accessible. We point out that the direct random analogue of the Duffin–Schaeffer conjecture, like the Duffin–Schaeffer conjecture itself, implies Catlin's conjecture (1976). It is not obvious whether the Duffin–Schaeffer conjecture and its random version imply one another, and it is not known whether Catlin's conjecture implies either of them. The question of whether Catlin's conjecture implies Duffin–Schaeffer conjecture has been unsettled for decades.

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Littlewood and Duffin–Schaeffer-Type Problems in Diophantine Approximation

Chow,

Technau

2024

Memoirs of the AMS

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Gallagher’s theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully inhomogeneous version of Gallagher’s theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin–Schaeffer conjecture for a class of nonmonotonic approximation functions.

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A note on the Duffin-Schaeffer conjecture with slow divergence

Cited by 15 publications

References 10 publications

The Duffin–schaeffer‐type Conjectures in Various Local Fields

The Duffin–schaeffer‐type Conjectures in Various Local Fields

Khintchine's Theorem With Random Fractions

Littlewood and Duffin–Schaeffer-Type Problems in Diophantine Approximation

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