Duffin Schaeffer Conjecture Research Articles

The Duffin–Schaeffer conjecture for systems of linear forms

AbstractWe extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no ‐by‐ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When , this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where and , in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the cases of that. Catlin's classical conjecture, where , follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where and , follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.

Littlewood and Duffin–Schaeffer-Type Problems in Diophantine Approximation

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.

The $p$-adic Duffin-Schaeffer conjecture

We prove Haynes' version of the Duffin-Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.

On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos–Maynard theorem

Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$, provided that the series $\sum _{q=1}^\infty \varphi (q) \psi (q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $\alpha$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is of asymptotic order $\sum _{q=1}^Q 2 \varphi (q) \psi (q) / q$. 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.

Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in ${\mathbb {R}}^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine–Groshev theorem that does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani [11] that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresneich et al. [5], and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin–Schaeffer conjecture. When $nm=1$, it is known by work of Duffin and Schaeffer [16] that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine–Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.

Khintchine’s theorem with rationals coming from neighborhoods in different places

The Duffin–Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine’s theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine’s theorem studied. We prove versions of Khintchine’s theorem where the rational numbers are sourced from a ball in some completion of [Formula: see text] (i.e. Euclidean or [Formula: see text]-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník’s theorem.

Difference Sets and the Metric Theory of Small Gaps

Abstract Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper, Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \mod 1, ~1 \leq n \leq N\}$ as $N \to \infty $, valid for Lebesgue-almost all $\alpha $ and formulated in terms of the additive energy of $\{a_1, \dots , a_N\}$. In the present paper, we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots , a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap and prove (slightly weaker) general upper and lower bounds that allow for a direct application. A major input for these results comes from the recent proof of the Duffin–Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick, and Radziwiłł on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

Variants of Khintchine's theorem in metric Diophantine approximation

New results towards the Duffin-Schaeffer conjecture, which is a fundamental problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function ψ:N→R we denote by W(ψ) the set of all x∈R such that |nx−a|<ψ(n) for infinitely many a,n. Analogously, denote W′(ψ) if we additionally require a,n to be coprime. Aistleitner et al. [2] proved that W′(ψ) is of full Lebesgue measure if there exist an ε>0 such that ∑n=2∞ψ(n)φ(n)/(n(log⁡n)ε)=∞. This result seems to be the best one can expect from the method used. Assuming the extra divergence ∑n=2∞ψ(n)/(log⁡n)ε=∞ we prove that W(ψ) is of full measure. This could also be deduced from the result in [2], but we believe that our proof is of independent interest, since its method is totally different from the one in [2]. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of ψ.

KHINTCHINE'S THEOREM WITH RANDOM FRACTIONS

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 implies Duffin-Schaeffer has been unsettled for decades.

The Duffin-Schaeffer conjecture with extra divergence

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function ψ:N→R for almost all reals x there are infinitely many coprime solutions (a,n) to the inequality |nx−a|<ψ(n), provided that the series ∑n=1∞ψ(n)φ(n)/n is divergent. In the present paper we prove that the conjecture is true under the “extra divergence” assumption that divergence of the series still holds when ψ(n) is replaced by ψ(n)/(log⁡n)ε for some ε>0. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.

JARNÍK’S THEOREM WITHOUT THE MONOTONICITY ON THE APPROXIMATING FUNCTION

Let [Formula: see text] be a non-negative function such that [Formula: see text] as [Formula: see text]. The well-known Jarník–Besicovtich theorem concerns the Hausdorff dimension of the set of [Formula: see text]- approximable numbers. In this paper, we give an alternative but short proof of the Jarník–Besicovitch theorem for approximating functions with no monotonicity. The main tool is the appropriate usage of the mass transference principle of Beresnevich–Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164(3) (2006) 971–992].

An analogue of Khintchine's theorem for self-conformal sets

AbstractKhintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin–Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz’s inhomogeneous (1958) version of Khintchine’s Theorem (1924). For example, given any real sequence [Formula: see text], we build a divergent series of non-negative reals [Formula: see text] such that for any [Formula: see text], almost no real number is inhomogeneously [Formula: see text]-approximable with inhomogeneous parameter [Formula: see text]. Furthermore, given any second sequence [Formula: see text] not intersecting the rational span of [Formula: see text], and assuming a dynamical version of Erdős’ Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously [Formula: see text]-approximable with any inhomogeneous parameter [Formula: see text]. Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin–Schaeffer Conjecture and Duffin–Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher’s Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.

DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

Given and , let denote the classical set of -approximable points in , which consists of that lie within distance from the lattice for infinitely many . In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold of and any almost all points on are not -approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set . In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for for any submanifold of and any satisfying . Here is the codimension of . We also show that the condition on is best possible and extend the result to general approximating functions.

THE DUFFIN–SCHAEFFER‐TYPE CONJECTURES IN VARIOUS LOCAL FIELDS

This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ if and only if $\sum_n\lambda({\mathcal E}_n)=\infty$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, \[ {\mathcal E}_n={\mathcal E}_n(\psi)=\bigcup_{m=1 \atop (m,n)=1}^n\big(\frac{m-\psi(n)}{n},\frac{m+\psi(n)}{n}\big), \] $\psi$ is any non-negative arithmetical function. Instead of studying $\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n$ we introduce an even fundamental object $\cup_{n=1}^{\infty}{\mathcal E}_n$ and conjecture there exists a universal constant $C>0$ such that \[\lambda(\bigcup_{n=1}^{\infty}{\mathcal E}_n)\geq C\min\{\sum_{n=1}^{\infty}\lambda({\mathcal E}_n),1\}.\] It 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 $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ 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.

A Contribution to Metric Diophantine Approximation: the Lebesgue and Hausdorff Theories

This thesis is concerned with the theory of Diophantine approximation from the point of view of measure theory. After the prolegomena which conclude with a number of conjectures set to understand better the distribution of rational points on algebraic planar curves, Chapter 1 provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is to develop a theory of approximation in the so far unstudied case when such curves are not defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities. In short, in this type of problem, one prescribes the set of integers which the denominators of all the possible rational approximants of a given vector have to belong to. Finally, a reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This provides the first example of a Khintchine type result in the context of so–called uniform approximation.

Approximation properties of β-expansions

Let be a real number. For a function , define to be the set of such that for infinitely many there exists a sequence satisfying . In Baker [Approximation properties of -expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every , the condition implies that is of full Lebesgue measure within . In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers satisfying , for Lebesgue almost every , the set is of full Lebesgue measure within . We also study the case where in which the set has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of .

Projective metric number theory

Abstract In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of ℚ. Using the projective metric studied in [Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 2, 211–248] we prove the analogue of Khintchine's theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin–Schaeffer conjecture.

A note on the Duffin-Schaeffer conjecture with slow divergence

For a non-negative function $\psi: ~ \N \mapsto \R$, let $W(\psi)$ denote the set of real numbers $x$ for which the inequality $|n x - a| < \psi(n)$ 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 {equation} \label{dsccond} \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n} = \infty. {equation} Recently Beresnevich, Harman, Haynes and Velani proved that $W(\psi)$ has full measure under the \emph{extra divergence} condition $$ \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n \exp(c (\log \log n) (\log \log \log n))} = \infty \qquad \textrm{for some $c>0$}. $$ In the present note we establish a \emph{slow divergence} counterpart of their result: $W(\psi)$ has full measure, provided\eqref{dsccond} holds and additionally there exists some $c>0$ such that $$ \sum_{n=2^{2^h}+1}^{2^{2^{h+1}}} \frac{\psi(n) \varphi(n)}{n} \leq \frac{c}{h} \qquad \textrm{for all \quad $h \geq 1$.} $$

The Duffin–Schaeffer conjecture with extra divergence II

Given a nonnegative function \({\psi\mathbb{N} \to \mathbb{R}}\), let W(ψ) denote the set of real numbers x such that |nx − a| 0). A consequence of our main result is that W(ψ) is of full Lebesgue measure if there exists an \({\epsilon > 0}\) such that $$ \textstyle \sum_{n\in\mathbb{N}}\left(\frac{\psi(n)}{n}\right)^{1+\epsilon}\varphi (n)=\infty. $$ The Duffin–Schaeffer Conjecture is the corresponding statement with \({\epsilon = 0}\) and represents a fundamental unsolved problem in metric number theory. Another consequence is that W(ψ) is of full Hausdorff dimension if the above sum with \({\epsilon = 0}\) diverges; i.e. the dimension analogue of the Duffin–Schaeffer Conjecture is true.