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Simon Kristensen

Publications and source records attributed to Simon Kristensen.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative Khintchine on the parabola with non-monotonic approximation functions

We prove a quantitative version of the convergence case of Khintchine's celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. A novel feature of our result is that unlike other results in literature, the approximating function is no longer required to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess' bound for character sums in short intervals.

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arXiv2026

Lower bounds for the Hausdorff dimension of expressible sets

We obtain positive lower bounds on the Hausdorff dimension of sets of real numbers given by expressions of the form $\sum_{n=1}^\infty \frac{1}{a_n b_n}$, where $b_n$ satisfies some growth condition and $a_n$ lies in some set, possibly depending on $n$. As a consequence of our results, some of the irrational numbers arising from Erd\H{o}s' celebrated construction from 1976 are not Liouville numbers.

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arXiv2025

Algebraic independence of infinite series

We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear independence of such series.

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arXiv2025

Infinite products with algebraic numbers

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{\alpha_{n,m}}\right)$, where the $\alpha_n$ and $\alpha_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities.

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arXiv2023

Asymptotics for partitions over the Fibonacci numbers and related sequences

In this paper, harkening back to ideas of Hardy and Ramanujan, Mahler and de Bruijn, with the addition of more recent results on the Fibonacci Dirichlet series, we determine the asymptotic number of ways $p_F(n)$ to write an integer as the sum of non-distinct Fibonacci numbers. This appears to be the first such asymptotic result concerning non-distinct partitions over Fibonacci numbers. As well, under weak conditions, we prove analogous results for a general linear recurrences.

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arXiv2023

A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture

In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teuli\'e. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $\tau(\phi^\omega(a))$ where $\phi^\omega(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|\phi^n(u)|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words.

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arXiv2022

BBP-type formulas -- an elementary approach

We provide a simple way of searching for formulas of the Bailey--Borwein--Plouffe type together with an algorithm and an implementation in \texttt{sage}. Aside from rediscovering some already known formulas, the method has been used in the discovery of a new BBP-type formula for $\sqrt{3}\pi$. In addition, the implementation is very flexible and allows us to look for BBP-type formulas to irrational bases but with integer coefficients. As an example of this, searching in various Pisot bases, we have discovered a formula for $\pi$ in base $1 + \sqrt{3}$, along with additional formulas.

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arXiv2021

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 Jarn\'ik and Lutz.

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arXiv2020

Metrical theorems on systems of affine forms

In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by $\qq\mapsto \qq X+\bfalpha$, where $\qq\in\Z^m$ (viewed as a row vector), $X$ is an $m\times n$ real matrix and $\bfalpha\in \R^n$. The classical setting refers to the ${\rm dist}(\qq X+\bfalpha, \Z^m)$ to measure the closeness of the integer values of the system $(X, \bfalpha)$ to integers. The absolute value setting is obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, \0)$; and the more general mixed settings are obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, Λ)$, where $Λ$ is a subgroup of $\Z^m$. We prove the Khintchine--Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

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arXiv2019

Irrationality and transcendence of continued fractions with algebraic integers

We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{α_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vertα_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $α= [0;α_1, α_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.

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arXiv2018

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

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arXiv2018

Metrical irrationality results related to values of the Riemann $ζ$-function

We introduce a one-parameter family of series associated to the Riemann $ζ$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost all is with respect to any sufficiently nice measure. We also give similar results for the Euler--Mascheroni constant, for $\sum_{n=1}^\infty \frac{1}{n^n}$ and for $\sum_{n=1}^\infty \frac{1}{n! +1}$. Finally, specialising the criteria used, we give some new criteria for the irrationality of $ζ(k)$, the Euler--Mascheroni constant and the latter two series.

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arXiv2016

Some remarks on Mahler's classification in higher dimension

We prove a number of results on the metric and non-metric theory of Diophantine approximation for Yu's multidimensional variant of Mahler's classification of transcendental numbers. Our results arise as applications of well known results in Diophantine approximation to the setting of Yu's classification.

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arXiv2013

A converse to linear independence criteria, valid almost everywhere

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.

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arXiv2012

Metrical musings on Littlewood and friends

We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teulié and a hybrid between a conjecture of Cassels and the Littlewood conjecture. It is shown that the set of numbers satisfying a strong version of all of these conjectures is large in the sense of Hausdorff dimension restricted to the set of badly approximable numbers.

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arXiv2011

Metrical results on systems of small linear forms

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine-Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximating function.

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