arXiv subjects

Manuel Hauke

Publications and source records attributed to Manuel Hauke.

29 records · Page 2Linked to original sources

arXiv2023

Quantitative inhomogeneous Diophantine approximation for systems of linear forms

The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the question considered by Duffin and Schaeffer for Khintchine's Theorem (which is the case $m = n = 1$), the question arises for which $m,n$ the monotonicity can be safely removed. If $m = n = 1$, it is known that monotonicity is needed. Recently, Allen and Ramirez showed that for $mn \geq 3$, the monotonicity assumption is unnecessary, conjecturing this to also hold when $mn = 2$. In this article, we confirm this conjecture for the case $(m,n)=(1,2)$ whenever the inhomogeneous parameter is a non-Liouville irrational number. Furthermore, under mild assumptions on the approximation function, we show an asymptotic formula (with almost square-root cancellation), which is not even known for homogeneous approximation. The proof makes use of refined overlap estimates in the 1-dimensional setting, which may have other applications including the inhomogeneous Duffin-Schaeffer conjecture.

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arXiv2023

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of $S_{N}(f, α) := \sum_{n=1}^N f(nα)$ for almost every $α$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for $N$ in a set of positive upper density, the order of $S_{N}(f, α)$ is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy-Koksma inequality for such $f$, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel-Bernstein Theorem that allows very general modularity conditions.

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arXiv2023

On the distribution of partial quotients of reduced fractions with fixed denominator

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with $N$. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators $N$. We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauss-Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient.

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arXiv2023

On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational

Dolgpoyat and Sarig showed that for any piecewise smooth function $f: \mathbb{T} \to \mathbb{R}$ and almost every pair $(α,x_0) \in \mathbb{T} \times \mathbb{T}$, $S_N(f,α,x_0) := \sum_{n =1}^{N} f(nα+ x_0)$ fails to fulfill a temporal distributional limit theorem. In this article, we show that the two-dimensional average is in fact not needed: For almost every $α\in \mathbb{T}$ and all $x_0 \in \mathbb{T}$, $S_N(f,α,x_0)$ does not satisfy a temporal distributional limit theorem, regardless of centering and scaling. The obtained results additionally lead to progress in a question posed by Dolgopyat and Sarig.

math.DS↗

arXiv2022

On the asymptotic behaviour of Sudler products for badly approximable numbers

Given a badly approximable number $α$, we study the asymptotic behaviour of the Sudler product defined by $P_N(α) = \prod_{r=1}^N 2 | \sin πr α|$. 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 $7$ infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value $7$ is optimal, even when restricting $α$ to be a quadratic irrational, which gives a negative answer to a question of the latter authors.

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arXiv2022

Poissonian correlations of higher orders

We show that any sequence $(x_n)_{n \in \mathbb{N}} \subseteq [0,1]$ that has Poissonian correlations of $k$-th order is uniformly distributed, also providing a quantitative description of this phenomenon. Additionally, we extend connections between metric correlations and additive energy, already known for pair correlations, to higher orders. Furthermore, we examine how the property of Poissonian $k$-th correlations is reflected in the asymptotic size of the moments of the function $F(t,s,N) = #\{n\leq N : \|x_n - t\| \leq s/(2N) \},\, t\in [0,1]. $

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arXiv2022

A Signature-Based Gröbner Basis Algorithm with Tail-Reduced Reductors (M5GB)

Gröbner bases are an important tool in computational algebra and, especially in cryptography, often serve as a boilerplate for solving systems of polynomial equations. Research regarding (efficient) algorithms for computing Gröbner bases spans a large body of dedicated work that stretches over the last six decades. The pioneering work of Bruno Buchberger in 1965 can be considered as the blueprint for all subsequent Gröbner basis algorithms to date. Among the most efficient algorithms in this line of work are signature-based Gröbner basis algorithms, with the first of its kind published in the late 1990s by Jean-Charles Faugère under the name F5. In addition to signature-based approaches, Rusydi Makarim and Marc Stevens investigated a different direction to efficiently compute Gröbner bases, which they published in 2017 with their algorithm M4GB. The ideas behind M4GB and signature-based approaches are conceptually orthogonal to each other because each approach addresses a different source of inefficiency in Buchberger's initial algorithm by different means. We amalgamate those orthogonal ideas and devise a new Gröbner basis algorithm, called M5GB, that combines the concepts of both worlds. In that capacity, M5GB merges strong signature-criteria to eliminate redundant S-pairs with concepts for fast polynomial reductions borrowed from M4GB. We provide proofs of termination and correctness and a proof-of-concept implementation in C++ by means of the Mathic library. The comparison with a state-of-the-art signature-based Gröbner basis algorithm (implemented via the same library) validates our expectations of an overall faster runtime for quadratic overdefined polynomial systems that have been used in comparisons before in the literature and are also part of cryptanalytic challenges.

math.AC↗

arXiv2022

Metric density results for the value distribution of Sudler products

We study the value distribution of the Sudler product $P_N(α) := \prod_{n=1}^{N}\lvert2\sin(πn α)\rvert$ for Lebesgue-almost every irrational $α$. We show that for every non-decreasing function $ψ: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{ψ(k)} = \infty$, the set $\{N \in \mathbb{N}: \log P_N(α) \leq -ψ(\log N)\}$ has upper density $1$, which answers a question of Bence Borda. On the other hand, we prove that $\{N \in \mathbb{N}: \log P_N(α) \geq ψ(\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if $\liminf_{k \to \infty} ψ(k)/(k \log k) \geq C$ for some sufficiently large $C > 0$.

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arXiv2022

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

Let $ψ: \mathbb{N} \to [0,1/2]$ 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 $\sum_{q=1}^\infty φ(q) ψ(q) / q$ 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 $q \leq Q$, is of asymptotic order $\sum_{q=1}^Q 2 φ(q) ψ(q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate coming 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.

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arXiv2021

Weak Poissonian Correlations

We examine a property of sequences called Poissonian pair correlations with parameter $0\leqslant β\leqslant 1$ (abbreviated as $β$-PPC). We prove that when $β<1,$ the property of $β$-PPC, also known as weak Poissonian correlations, can be detected at the behaviour of sequences at small scales, and show that this does not happen for the classical notion of PPC, that is, when $β= 1$. Furthermore, we show that whenever $0\leqslant α< β\leqslant 1$, $β$-PPC is stronger than $α$-PPC. We also include a discussion on weak Poissonian correlations of higher orders, showing that for $β< 1$, Poissonian $β$-correlations of order $k+1$ imply Poissonian $β$-correlations of $k$-th order with the same parameter $β$.

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arXiv2021

On extreme values for the Sudler product of quadratic irrationals

Given a real number $α$ and a natural number $N$, the Sudler product is defined by $P_N(α) = \prod_{r=1}^{N} 2 \left\lvert \sin(π\left(rα\right))\right\rvert.$ Denoting by $F_n$ the $n$--th Fibonacci number and by $ϕ$ the Golden Ratio, we show that for $F_{n-1} \leq N < F_n$, we have $P_{F_{n-1}}(ϕ)\leq P_N(ϕ) \leq P_{F_{n}-1}(ϕ)$ and $\min_{N \geq 1} P_N(ϕ) = P_1(ϕ)$, thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for $\liminf_{N \to \infty} P_N(ϕ)$ and $\limsup_{N \to \infty} \frac{P_N(ϕ)}{N}$ whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals $β$ with continued fraction expansion $β= [0;b,b,b\ldots]$ where $1 \leq b \leq 5$, completing the calculation of $\liminf_{N \to \infty} P_N(β)$, $\limsup_{N \to \infty} \frac{P_N(β)}{N}$ for $β$ being an arbitrary quadratic irrational with continued fraction expansion of period length 1.

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