8 citations · 16 across the 13 of their papers we have counts for
10 papers · 1 filter
Curious congruences for cyclotomic polynomials
Shigeki Akiyama, Hajime Kaneko
Let be the -th derivative of -th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: is divisible by $ϕ(…
Exponential Diophantine approximation and symbolic dynamics
Shigeki Akiyama, Teturo Kamae, Hajime Kaneko
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 probl…
Products of integers with few nonzero digits
Hajime Kaneko, Thomas Stoll
Let be the number of nonzero bits in the binary digital expansion of the integer . We study, for fixed , the Diophantine system $$ s(ab)=k, \quad s(a)=\ell,\qua…
Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers
Shigeki Akiyama, Hajime Kaneko
Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe…
The digit exchanges in the rotational beta expansions of algebraic numbers
Hajime Kaneko, Makoto Kawashima
In this article, we investigate the -expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where is a P…
Arithmetical properties of real numbers related to beta-expansions
Hajime Kaneko
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…
On subwords in the base- expansion of polynomial and exponential functions
Hajime Kaneko, Thomas Stoll
Let be any word over the alphabet , and denote by either a polynomial of degree or for a fixed . Furthermore, denote by…
On the digital representation of smooth numbers
Yann Bugeaud, Hajime Kaneko
Let be an integer. Among other results, we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of cannot have simultaneousl…
Bernoulli-Carlitz and Cauchy-Carlitz numbers with Stirling-Carlitz numbers
Hajime Kaneko, Takao Komatsu
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…
Cauchy-Carlitz numbers
Hajime Kaneko, Takao Komatsu
In 1935 Carlitz introduced Bernoulli-Carlitz numbers as analogues of Bernoulli numbers for the rational function field . In this paper, we introduce Cauchy-Carlitz…