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Manuel Hauke

Publications and source records attributed to Manuel Hauke.

At least 19 recordsLinked to original sources
arXiv2026

The bad and rough rotation is Poissonian

Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_nα)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N} $ and $α$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $α$, $(a_nα)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_nα)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $α$, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod $d$ in diophantine Bohr sets which might be of independent interest.

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arXiv2026

Proving the Duffin-Schaeffer conjecture without GCD graphs

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the $(\log Ψ(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log Ψ(N))^{\frac{1}{2} - \varepsilon})$.

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arXiv2026

Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences $(q_n)_{n\in\mathbb{N}}$, we show that the covering radius $\frac{1}{n}$ is sharp up to a constant: there exist $C>c>0$ such that, for Lebesgue-almost all $x$, the intervals of length $\frac{C}{n}$ cover $[0,1)$ infinitely often, while this fails for intervals of length $\frac{c}{n}$. Moreover, the lower bound holds for certain sub-lacunary rates and the results partially extend to all probability measures with sufficiently fast Fourier decay. As an application, we obtain a new bound for a variant of the inhomogeneous Littlewood-Cassels problem: for any badly approximable $α$ and $γ\in\mathbb{R}$, there exists a set of badly approximable $β$ of full Hausdorff dimension such that $ \|nα-γ\| \|nβ-δ\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $δ\in\mathbb{R}$. This improves upon previous works of Haynes-Jensen-Kristensen, Chow-Technau, and the third author, and is best possible when one restricts to best approximations of the first factor. Second, under certain arithmetic restrictions on $(q_n)_{n\in\mathbb{N}}$, we compute the almost-sure Hausdorff dimension of limsup sets generated by intervals of size $\frac{1}{n^ν}$ for $ν\geqslant 1$, centered at $\{q_n x\}$, and intersected with Ahlfors regular compact sets such as the middle-third Cantor set. In particular, our results apply to all real-valued lacunary sequences, to integer-valued polynomials, and to powers of primes. This substantially extends the work of Bugeaud and Durand, which applies only to certain super-lacunary integer-valued sequences.

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arXiv2026

The Prime times of twisted Diophantine approximation

The seminal work of Kurzweil (1955) provides for any fixed badly approximable $α$ and monotonically decreasing $ψ$ a Khintchine-type statement on the set of the inhomogeneous real parameters $γ$ for which $\lVert n α+ γ\rVert \leq ψ(n)$ has infinitely many integer solutions, and further shows that the assumption of $α$ being badly approximable is necessary. In this article, we generalize Kurzweil's statement to restricting $n \in \mathcal{A}$, where $\mathcal{A} \subseteq \mathbb{N}$ is a set with some multiplicative structure. We show that for badly approximable $α$, the result of Kurzweil extends to a general class of sets $\mathcal{A}$, which allows us to establish the Kurzweil-type result in particular along the primes and along the sums of two squares. Furthermore, we construct non-trivial sets $\mathcal{A}$ where the assumption of $α$ being badly approximable is necessary. In particular, this criterion applies to $\mathcal{A}$ being the set of square-free numbers, providing a novel characterization of the badly approximable numbers. These statements in particular allow for improving the best known bounds for $\lVert n α+ γ\rVert \leq ψ(n)$ for infinitely many $n \in \mathcal{A}$ for fixed badly approximable $α$ and for various sets $\mathcal{A}$ of number-theoretic interest when accepting an exceptional set for $γ$ of Lebesgue measure $0$.

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arXiv2025

Rational approximation with chosen numerators

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.

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arXiv2025

On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations

The colored Jones polynomial $J_{K,N}$ is an important quantum knot invariant in low-dimensional topology. In his seminal paper on quantum modular forms, Zagier predicted the behavior of $J_{K,0}(e^{2 πi x})$ under the action of $SL_2(\mathbb{Z})$ on $x \in \mathbb{Q}$. More precisely, Zagier made a prediction on the asymptotic value of the quotient $J_{K,0}(e^{2 πi γ(x)})/ J_{K,0}(e^{2 πi x})$ for fixed $γ\in SL_2(\mathbb{Z})$, as $x \to \infty$ along rationals with bounded denominator. In the case of the figure-eight knot $4_1$, which is the most accessible case, there is an explicit formula for $J_{4_1,0}(e^{2 πi x})$ as a sum of certain trigonometric products called Sudler products. By periodicity, the behavior of $J_{4_1,0}(e^{2 πi x})$ under the mapping $x \mapsto x+1$ is trivial. For the second generator of $SL_2(\mathbb{Z})$, Zagier conjectured that with respect to the mapping $x \mapsto 1/x$, the quotient $h(x) = \log ( J_{4_1,0}(e^{2 πi x}) / J_{4_1,0}(e^{2 πi /x}))$ can be extended to a function on $\mathbb{R}$ that is continuous at all irrationals. This conjecture was recently established by Aistleitner and Borda in the case of all irrationals that have an unbounded sequence of partial quotients in their continued fraction expansion. In the present paper we prove Zagier's continuity conjecture in full generality.

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arXiv2025

Twisted approximation with restricted denominators

Given an increasing integer sequence $(a_n)$, a real number $α$, and a sequence $ψ(n)$, we study the set $W$ of real numbers $γ$ for which $a_nα- γ$ is a distance less than $ψ(n)$ away from an integer. This is often referred to as twisted Diophantine approximation, in this case with denominators restricted to the given sequence $(a_n)$. Our main results are about the size of $W$, and they hold for almost every $α$, with respect to a measure of positive Fourier dimension, for example Lebesgue measure. Our results extend recent work of Kristensen and Persson, and answer questions that they posed.

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arXiv2025

Moment generating functions and moderate deviation principles for lacunary trigonometric sums

In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 πn_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard gap condition $n_{k+1} / n_k \geq q > 1$ for $k \geq 1$. A crucial ingredient in their work were asymptotic estimates for the moment generating function (MGF) of such sums, which turned out to depend on the fine arithmetic structure of the sequence $(n_k)_{k \geq 1}$ in an intricate way. In the present paper we carry out a detailed study of the MGF for lacunary trigonometric sums (without any structural assumptions on the underlying sequence, other than lacunarity), and we determine the sharp threshold where arithmetic effects start to play a role. As an application, we prove moderate deviation principles for lacunary trigonometric sums, and show that the tail probabilities are in accordance with Gaussian behavior throughout the whole range between the central limit theorem and the LDP regime.

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arXiv2025

A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard

In this note, we review the history of Khintchine's Theorem which is the foundation of metric Diophantine approximation, and discuss several generalizations and recent breakthroughs in this area. We focus particularly on the direction of the Duffin-Schaeffer Conjecture which was spectacularly proven in 2020. We present some simplified key ideas of the proof that can also be applied in various other areas of number theory.

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arXiv2025

General Duffin--Schaeffer-type counterexamples in diophantine approximation

Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erdős, Vaaler, and Yu.

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arXiv2025

Limit laws for cotangent and Diophantine sums

Limit laws for ergodic averages with a power singularity over circle rotations were first proved by Sinai and Ulcigrai, as well as Dolgopyat and Fayad. In this paper, we prove limit laws with an estimate for the rate of convergence for the sum $\sum_{n=1}^N f(n α)/n^p$ in terms of a $1$-periodic function $f$ with a power singularity of order $p \ge 1$ at integers. Our results apply in particular to cotangent sums related to Dedekind sums, and to sums of reciprocals of fractional parts, which appear in multiplicative Diophantine approximation. The main tools are Schmidt's method in metric Diophantine approximation, the Gauss-Kuzmin problem and the theory of $ψ$-mixing random variables.

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arXiv2025

On a Problem of Kac concerning Anisotropic Lacunary Sums

Given a lacunary sequence $(n_k)_{k \in \mathbb{N}}$, arbitrary positive weights $(c_k)_{k \in \mathbb{N}}$ that satisfy a Lindeberg-Feller condition, and a function $f: \mathbb{T} \to \mathbb{R}$ whose Fourier coefficients $\hat{f_k}$ decay at rate $\frac{1}{k^{1/2 + \varepsilon}}$, we prove central limit theorems for $\sum_{k \leq N}c_kf(n_kx)$, provided $(n_k)_{k \in \mathbb{N}}$ satisfies a Diophantine condition that is necessary in general. This addresses a question raised by M. Kac [Ann. of Math., 1946].

math.PR↗
arXiv2024

The Duffin-Schaeffer conjecture with a moving target

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (φ(q)ψ(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<ψ(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprindžuk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter $\mathbf{y}$ is free to vary with $q$. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets.

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arXiv2024

Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer

The most versatile version of the classical divergence Borel-Cantelli lemma shows that for any divergent sequence of events $E_n$ in a probability space satisfying a quasi-independence condition, its corresponding limsup set $E_\infty$ has positive probability. In particular, it provides a lower bound on the probability of $E_\infty$. In this paper we establish a new version of this classical result which guarantees, under an additional mild assumption, that the probability of $E_\infty$ is not just positive but is one. Unlike existing optimal results, it is applicable within the setting of arbitrary probability spaces. We then go onto to consider a range of applications in number theory and dynamical systems. These include new results on the inhomogeneous Duffin-Schaeffer conjecture. In particular, we establish alternatives to the classical (homogeneous) zero-one laws of Cassels and Gallagher and use them to resolve the so-called weak Duffin-Schaeffer conjecture for an arbitrary rational inhomogeneous shift. As a bi-product, we establish the Duffin-Schaeffer conjecture with congruence relations. The applications to dynamical systems include new characterisations of Borel-Cantelli sequences and new dynamical Borel-Cantelli lemmas, as well as characterising Khintchine-type sequences for shrinking targets.

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arXiv2024

Hausdorff dimension estimates for Sudler products with positive lower bound

Given an irrational number $α$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(α) = \prod_{r=1}^N 2\lvert \sin πr α\rvert$. We show that $\liminf_{N \to \infty} P_N(α) >0$ and $\limsup_{N \to \infty} P_N(α)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $α$ that satisfy $\limsup_{N \to \infty} P_N(α)/N < \infty,\liminf_{N \to \infty} P_N(α) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of such $α$ is invariant under the Gauss map $T$.

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arXiv2024

On Inhomogeneous Poissonian Pair Correlations

We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair correlations need not be uniformly distributed, contrary to what was till recently believed.

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arXiv2024

The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Given a monotonically decreasing $ψ: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $α\in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that $\left\lvert α- \frac{p}{q}\right\rvert \leq \frac{ψ(q)}{q}$. The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any $k\geq 1$, any function $ψ: \mathbb{N} \to [0,1/2]$ (not necessarily monotonic) and almost every $α\in \mathbb{R}^k$, there exist infinitely many $q$ such that $\prod\limits_{i=1}^k \left\lvert α_i - \frac{p_i}{q}\right\rvert \leq \frac{ψ(q)}{q^k}, p_1,\ldots,p_k$ all coprime to $q$, if and only if \[\sum\limits_{q \in \mathbb{N}} ψ(q) \left(\frac{φ(q)}{q} \right)^k\log \left(\frac{q}{φ(q)ψ(q)}\right)^{k-1} = \infty.\] This settles a conjecture of Beresnevich, Haynes, and Velani.

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