arXiv subjects

Hajime Kaneko

Publications and source records attributed to Hajime Kaneko.

At least 19 recordsLinked to original sources

arXiv2026

An extension of algebraic independence of special values for non-lacunary power series

We study the algebraic independence of special values of power series $f(β^{-1})$, where $β$ is a fixed Pisot or Salem number. In particular, we consider the case where $f(X)=\sum_{n\geq 0} t(n) X^{w(n)}$ is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: \[ % \begin{gathered} \sum_{n=3}^{\infty}\lfloor n^{y}\rfloorβ^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=3}^{\infty}β^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=1}^{\infty}β^{-\lfloor n^{\log n}\rfloor}, % \end{gathered} \] where $y$ is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients $(t(n))_{n\geq 0}$ and the exponents $(w(n))_{n\geq 0}$.

math.NT↗

arXiv2026

Rotational beta expansions and Schmidt games

We consider rotational beta expansions in dimensions 1, 2 and 4 and view them as expansions on real numbers, complex numbers, and quaternions, respectively. We give sufficient conditions on the parameters $α, β\in (0,1)$ so that particular cylinder sets arising from the expansions are winning or losing Schmidt $(α,β)$-game.

math.NT↗

arXiv2026

On the irrationality exponent of real numbers with low complexity expansion

Let $ξ$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $ξ$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $ξ$, where $p(n, \mathbf{x})$ counts the number of distinct blocks of length $n$ in $\mathbf{x}$, for $n \ge 1$. If the irrationality exponent of $ξ$ is equal to $2$, which is the case for almost all real numbers $ξ$, we show that the limit superior of the sequence $(p(n, \mathbf{x}) / n)_{n \ge 1}$ is at least equal to 4/3. The proof is based on a careful study of the evolution of the Rauzy graphs of infinite words of low complexity.

math.NT↗

arXiv2026

Refinements of Erdős's irrationality criterion for certain sparse infinite series

In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erdős and obtain several irrationality results for various infinite series involving the classical arithmetic functions. For example, we prove that for any integers $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{σ(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{ϕ(n)}} \] are both irrational, where $d(n)$, $σ(n)$, and $ϕ(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively.

math.NT↗

arXiv2025

Borel Complexity of the set of vectors normal for a fixed recurrence sequence

In this paper, we consider recurrence sequences $x_n=ξ_1 α_1^n+ξ_2 α_2^n$ ($n=0,1,\ldots$) with companion polynomial $P(X)$. For example, the sequence $x_n=ξ_1(4+\sqrt{2})^n+ξ_2(4-\sqrt{2})^n$ satisfies the recurrence $x_{n+2}-8x_{n+1}+14x_n=0$ and has companion polynomial $P(X)=X^2-8X+14=(X-4-\sqrt{2})(X-4+\sqrt{2})$. We call $(ξ_1,ξ_2)$ normal with respect to the recurrence relation determined by $P(X)$ when $(x_n)_{n\ge 0}$ is uniformly distributed modulo one. Determining the Borel complexity of the set of normal vectors for a fixed recurrence sequence is unresolved even for most geometric progressions. Under certain assumptions, we prove that the set of normal vectors is $\boldsymbolΠ_3^0$-complete. A special case is the new result that the sets of numbers normal in base $α$, i.e. $\{ξ\in \mathbb{R}\mid (ξα^n)_{n\geq 0}\mbox{ is u.d. modulo one.} \}$, are $\boldsymbolΠ_3^0$-complete for every real number $α$ with $|α|$ Pisot. We analyze the fractional parts of recurrence sequences in terms of finite words via certain numeration systems. One of the difficulties in proving the main result is that even when recurrence sequences are uniformly distributed modulo one, it is not known what the average frequencies of the digits in the corresponding digital expansions are or if they even must exist.

math.LO↗

arXiv2025

Exponential Diophantine approximation and symbolic dynamics

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected on the Markoff-Lagrange spectrum.

math.NT↗

arXiv2023

Markoff-Lagrange spectrum of one-sided shifts

For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are $S$-adic sequences, where $S$ is a certain infinite set of substitutions that includes Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff-Lagrange spectrum.

math.DS↗

arXiv2022

Curious congruences for cyclotomic polynomials

Let $Φ_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2Φ^{(3)}_n(1)$ is divisible by $ϕ(n)-2$ and $Φ^{(2k+1)}_n(1)$ is divisible by $ϕ(n)-2k$ for $k\ge 2$. The congruence stems from a general property of self-reciprocal polynomials.

math.NT↗

arXiv2022

On the binary digits of $n$ and $n^2$

Let $s(n)$ denote the sum of digits in the binary expansion of the integer $n$. Hare, Laishram and Stoll (2011) studied the number of odd integers such that $s(n)=s(n^2)=k$, for a given integer $k\geq 1$. The remaining cases that could not be treated by theses authors were $k\in\{9,10,11,14,15\}$. In this paper we show that there is only a finite number of solutions for $k\in\{9,10,11\}$ and comment on the difficulties to settle the two remaining cases $k\in\{14,15\}$. A related problem is to study the solutions of $s(n^2)=4$ for odd integers. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that $n=13,15,47,111$ are the only solutions. In this paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for $s(n^2)=5$.

math.NT↗

arXiv2022

The digit exchanges in the rotational beta expansions of algebraic numbers

In this article, we investigate the $β$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $β$ is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers. Our method is applicable also to the digit expansions of complex algebraic numbers, which gives new estimation. In particular, we investigate the digits of rotational beta expansion by Akiyama and Caalim $[3]$ and zeta-expansion by Surer $[20]$, where the base is a quasi-Pisot or quasi-Salem number.

math.NT↗

arXiv2021

Products of integers with few nonzero digits

Let $s(n)$ be the number of nonzero bits in the binary digital expansion of the integer $n$. We study, for fixed $k,\ell,m$, the Diophantine system $$ s(ab)=k, \quad s(a)=\ell,\quad \mbox{and }\quad s(b)=m, $$ in odd integer variables $a,b$. When $k=2$ or $k=3$, we establish a bound on $ab$ in terms of $\ell$ and $m$. While such a bound does not exist in the case of $k=4$, we give an upper bound for $\min\{a,b\}$ in terms of $\ell$ and $m$.

math.NT↗

arXiv2021

Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal L}(α)=\left\{\left.\limsup_{n\to \infty}\|ξα^n\|\ \right|\ ξ\in {\mathbb R}\right\}, \] where $\|x\|$ is the distance from $x$ to the nearest integer. First, we show that ${\mathcal L}(α)$ is closed in $[0,1/2]$ for any Pisot number $α$. Then we consider the case where $α$ is an integer with $α\geq 2$, or a quadratic unit with $α\ge 3$. We show that ${\mathcal L}(α)$ contains a proper interval when $α$ is quadratic but it does not when $α$ is an integer. We also determine the minimum limit point and all isolated points beneath this point. In the course of the proof, we revisit a property studied by Markoff which characterizes bi-infinite balanced words and sturmian words.

math.NT↗

arXiv2019

Generic point equivalence and Pisot numbers

Let $β>1$ be an integer or generally a Pisot number. Put $T(x) = \{ βx \}$ on $[0,1]$ and let $S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm β^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.

math.DS↗

arXiv2019

Bernoulli-Carlitz and Cauchy-Carlitz numbers with Stirling-Carlitz numbers

Recently, the Cauchy-Carlitz number was defined as the counterpart of the Bernoulli-Carlitz number. Both numbers can be expressed explicitly in terms of so-called Stirling-Carlitz numbers. In this paper, we study the second analogue of Stirling-Carlitz numbers and give some general formulae, including Bernoulli and Cauchy numbers in formal power series with complex coefficients, and Bernoulli-Carlitz and Cauchy-Carlitz numbers in function fields. We also give some applications of Hasse-Teichmüller derivative to hypergeometric Bernoulli and Cauchy numbers in terms of associated Stirling numbers.

math.NT↗

arXiv2018

Corona limits of tilings : Periodic case

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron (see [Zhuravlev 2001], [Maleev-Shutov 2011]).

math.MG↗

arXiv2018

Hensel's lemma for general continuous functions

In the present paper, we generalize the well-known Hensel's lifting lemma to any continuous function $f : \mathbb{Z}_p\rightarrow \mathbb{Z}_p$. This answers a question posed by Axelsson and Khrennikov (2016) who showed the validity of Hensel's lemma for $1$- and for $p^α$-Lipschitz functions. For the statement and the proof, we introduce a suitable generalization of the original van der Put series. We use the concept of approximability of continuous functions to give numerical examples.

math.AC↗

arXiv2017

Arithmetical properties of real numbers related to beta-expansions

The main purpose of this paper is to study the arithmetical properties of values $\sum_{m=0}^{\infty} β^{-w(m)}$, where $β$ is a fixed Pisot or Salem number and $w(m)$ ($m=0,1,\ldots$) are distinct sequences of nonnegative integers with $w(m+1)>w(m)$ for any sufficiently large $m$. We first introduce criteria for the algebraic independence of such values. Our criteria are applicable to certain sequences $w(m)$ ($m=0,1,\ldots$) with $\lim_{m\to\infty}w(m+1)/w(m)=1.$ For example, we prove that two numbers \[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{-\lfloor φ(0,1;m)\rfloor}\] are algebraically independent, where $φ(1,0;m)=m^{\log m}$ and $φ(0,1;m)=m^{\log\log m}$. \par Moreover, we also give criteria for linear independence of real numbers. Our criteria are applicable to the values $\sum_{m=0}^{\infty}β^{-\lfloor m^ρ\rfloor}$, where $β$ is a Pisot or Salem number and $ρ$ is a real number greater than 1.

math.NT↗

arXiv2017

On subwords in the base-$q$ expansion of polynomial and exponential functions

Let $w$ be any word over the alphabet $\{0,1,\ldots, q-1\}$, and denote by $h$ either a polynomial of degree $d\geq 1$ or $h: n\mapsto m^n$ for a fixed $m$. Furthermore, denote by $e_q(w;h(n))$ the number of occurrences of $w$ as a subword in the base-$q$ expansion of $h(n)$. We show that \[ \limsup_{n\to\infty} \frac{e_q(w;h(n))}{\log n}\geq \frac{γ(w)}{l\log q}, \] where $l$ is the length of $w$ and $γ(w)\geq 1$ is a constant depending on a property of circular shifts of $w$. This generalizes work by the second author as well as is related to a generalization of Lagarias of a problem of Erdős.

math.NT↗