Workshop “Number Theory and Ergodic Theory”(February 10 – 12, 2024 at Kanazawa University Satellite Plaza)

Abstracts of talks
Hiroshi Fujisaki (Kanazawa University)
Title: On topological entropies of the subshifts associated with the stream version of asymmetric binary systems
Abstract: The stream version of asymmetric binary systems (ABS) invented by Duda is anentropy coder for information sources with a ﬁnite alphabet. It has the state parameter l ofa nonnegative integer and the probability parameter p with 0 < p < 1. First we observethat the edge shift XG associated with the stream version of ABS has the topological entropyh(XG ) = log 2. Then we deﬁne the edge shift XH associated with output blocks from the stream
version of ABS, and show that h(XH ) = h(XG ). The encoding and decoding algorithms for thestream version of ABS establish a bijection√ between XG and XH . We consider the case wherep = 1/β with the golden mean β = (1 + 5)/2, and show that XG and XH are conjugate forl = 7.
Keywords: topological entropy, shift space, edge shift, conjugacy, asymmetric binary systems(ABS)Hajime Kaneko (University of Tsukuba)
Title: Analogue of Markoﬀ and Lagrange spectra on one-sided shift spaces
Abstract: Recently, multiplicative analogue of Markoﬀ-Lagrange spectrum was investigated,which is related to the fractional parts of geometric progressions. This analogy is investigated
via limit sup words on shift spaces. In particular, Dubickas showed that the minimal limit
points of certain spectra are denoted in terms of a substitution. In this talk, we consider generalordered shift spaces and analogue of Markoﬀ and Lagrange spectra. In our main results, wedetermine the minimal limit points and discrete parts. In particular, we show that the minimallimit points are denoted in terms of S-adic sequences, where S is an inﬁnite set of substitutions.
This is a joint work with Wolfgang Steiner.
Keywords: Markoﬀ and Lagrange spectra, discrete part, S-adic sequences
Yasushi Nagai (Shinshu University)
Title: Overlap algorithm for general S-adic tilings
Abstract: We investigate the question of when a tiling has pure point spectrum, for the classof S-adic tilings, which includes all self-aﬃne tilings. The overlap algorithm by Solomyak isa powerful tool to study this problem for the class of self-aﬃne tilings. We generalize thisalgorithm for general S-adic tilings, and apply it to a class of block S-adic tilings to show almost
all of them have pure point spectra. This is a joint work with Jorg Thuswaldner.
Keywords: tiling, pure point spectrum, overlap algorithm
Hideki Tsuiki (Kyoto University)
Title: TriMata and Tiling
Abstract: 3D TriMata is an assembly kit composed of pieces shaped as two connected hexahedra, derived by dividing a regular regular tetrahedron into four identical shapes. This kit
enables the construction of models based on the Sierpinski tetrahedron. Based on this fact, weshow two 3D tiles: one allows for periodical tiling, but when both tiles are required to be used,then only aperiodic tiling is possible.
Keywords: aperiodic tiling, Sierpinski tetrahedron
Tadahisa Hamada (University of Tsukuba)
Title: Transition process of factor graphs of Sturmian words
Abstract: This study elucidates the properties of Sturmian words by using the graph repre1

sentation of factors, known as factor graph, which was introduced by Rauzy (1982).
Keywords: Sturmian word, factor graph
Teturo Kamae (Osaka Metropolitan University)
Title: Inﬁnite self-shuﬄing words
Abstract: An inﬁnite word x = x0 x1 x2 · · · over an alphabet A is called self-shuﬄing if there
exists a partition N1 ∪N2 = {0, 1, 2, · · · } such that both x restricted to N1 and N2 are x. An easy
example is Fibonacci word x which is invariant under the substitution ϕ : 0 → 01, 1 → 0, sincein this case we have x = ϕ2 (x) = ϕ(x0 )x0 ϕ(x1 )x1 · · · , and hence ϕ(x0 )ϕ(x1 ) · · · = x0 x1 · · · = x.
We prove a necessary and suﬃcient condition to be self-shuﬄing and prove that the Thue-Morseword and any Sturmian word with non-zero intercept are self-shuﬄing. This is a joint work with
Emilie Charlier, Svetlana Puzynina and Luca Zamboni.
Keywords: self-shuﬄing words, Thue-Morse word, Sturmian words
Yohei Tachiya (Hirosaki University)
Title: Linear independence of inﬁnite series related to the Thue-Morse sequence along powers
Abstract: The Thue-Morse sequence along the powers has been studied by several authors. In
this talk, we give a linear independence result for the values of inﬁnite series generated by thesesequences. This generalizes an irrationality result of E. Miyanohora(2023) who treated the caseof the squares. This is a joint work with Michael Coons.
Keywords: Thue-Morse sequence, Linear independence, Pisot numbers, Salem numbers
Masamichi Yoshida (Osaka Metropolitan University)
Title: Signed Expansion and associated Odometer based on Cubic Pisot Unit (Joint work withFumichika Takamizo)
Abstract: First signed expansion of integers based on a cubic Pisot unit is introduced. Then
we study the associated odometer and domain exchange.
Keywords: Signed expansion, odometer, domain exchange
Nathaniel Nollen (University of the Philippines-Diliman)
Title: Number of Nonzero Digits in Quaternion Expansions
Abstract: Let {η1 , η2 , η3 , η4 } be an R-basis of the skew-ﬁeld H of real quaternions. We consider
the lattice L := Zη1 + Zη2 + Zη3 + Zη4 and the fundamental domainχ := {t1 η1 + t2 η2 + t3 η3 + t4 η4 : t1 , t2 , t3 , t4 ∈ [0, 1)}.
For q ∈ H with |q| > 1, deﬁne the transformation Tq on χ by Tq (z) = qz − d(z) where d(z) isthe unique element d of L such that qz − d ∈ χ.
The quaternion expansion of z ∈ χ in base q is the sumz = q −1 d1 + q −2 d2 + q −3 d3 + · · · ,where dj = d(T j−1 (z)). We identify this expansion with the sequence (d1 , d2 , d3 , . . . ) of digits
di ’s. In this presentation, we provide some results on the number of nonzero digits in quaternion
expansions as a measure of digit complexity.
Keywords: quaternion, expansions, fundamental domain, lattice, skew ﬁeldAsaki Saito (Future University Hakodate)
Title: Exact binary expansion of real algebraic integers using Newton’s method
Abstract: We introduce a method for computing exact binary expansions of real algebraic
integers using Newton’s method. This method allows us to speed up the true orbit pseudorandom
number generation that we have proposed so far. This is a joint work with Akihiro Yamaguchi
of Fukuoka Institute of Technology.
Keywords: Newton’s method, algebraic integer, binary expansion, pseudorandom number2

Mako Toyosumi (Japan Women’s University)
Title: Some complex continued fractions with coeﬃcients in a submodule
Abstract: We discuss a complex analogue of continued fractions of even integers in the realcase. This type of complex continued fractions is ﬁrst considered by J. Hurwitz for the Gaussianﬁeld. In this talk, we consider the case of the Eisenstein ﬁeld. We also consider the analogue ofthe odd integer case.
Keywords: complex continued fraction
Dong Han Kim (Dongguk University - Seoul)
Title: Continued fraction algorithm for rational numbers of certain parity types
Abstract: There are three diﬀerent kinds of rational numbers according to the parity of thenumerator and denominator, say odd/odd, even/odd and odd/even rational numbers. For agiven irrational number, we consider the properties of best rational approximations of givenparities. We study continued fraction algorithms to ﬁnd best rational approximations by rational
numbers of given parities and compare the algorithms with continued fraction expansions. Thisis joint work with Seul Bee Lee and Lingmin Liao.
Keywords: Continued fraction, intermediate convergents, Diophantine approximation
Shin-ichi Yasutomi (Toho University)
Title: Simultaneous convergent continued fraction algorithm with two places and the potentialof ergodic properity.
Abstract: Let p be a prime number and K be a ﬁeld with embeddings into R and Qp . We
propose an algorithm that generates continued fraction expansions converging in Qp and isexpected to simultaneously converge in both R and Qp . This algorithm produces ﬁnite continued
fraction expansions for rational numbers. In the case of p = 2 and if K is a quadratic ﬁeld, thecontinued fraction expansions generated by this algorithm converge in R, and they are eventuallyperiodic or ﬁnite. For an element α in K, let pn /qn denote the n-th convergent. There exist
constants u1 and u2 in R>0 with u1 + u2 = 2, and constants C1 and C2 in R>0 such that|α − pn /qn | < C1 /|qn |u1 and |α − pn /qn |2 < C2 /|qn |u2 . Here, | · |2 represents the 2-adic distance.
For prime numbers p > 2, we present numerical experiences. Furthermore, this algorithm has
the potential for natural extension, with mention of its ergodic features being captured.
Keywords: continued fraction, simultaneous approximation
Haruyoshi Tanaka (Naruto University of Education)
Title: Higher-order asymptotic behaviours of pressure functionals and statistical representationsof the coeﬃcients
Abstract: We study the higher-order asymptotic behaviour of the pressure functional ϵ 7→P (φ+ϵψ) with potentials φ and ψ on a countable Markov shift X. We show that if the transition
matrix of X is ﬁnitely primitive, the potentials φ and ψ are real-valued locally Hölder continuousfunctions on X, and a suﬃcient condition for an asymptotic expansion of ϵ 7→ P (φ + ϵψ) issatisﬁed, then the 3-th coeﬃcient of this expansion has a limit representation which looks likethe asymptotic
variancewell. In this study, we also give an explicit formula for the behaviour∫∑m−1of µ(( i=0 (ψ − X ψ dµ) ◦ σ i )k ) as m → ∞ with the Gibbs measure µ for φ and the shifttransformation σ. In application, we apply the convergence rate of the mean ergodic theoremin Lk (µ) with even integer k to our result.
Keywords: asymptotic variance, topological pressure, countable Markov shift
Soonki Hong (Postech)
Title: Weak Ramanujan property of standard non-uniform arithmetic quotient for P GL3 andP GL4
Abstract: As a generalization of Morgenstern’s deﬁnition of the Ramanujan diagram, Samuelsdeﬁned Ramanujan complexes on the quotient of the Bruhat-Tits building Bd associated toP GLd (Fq ((t−1 ))) by a congruence subgroup of P GLd (Fq [t]). She proved that if d is prime, the3quotient is not Ramanujan but is weak Ramanujan. In this talk, we will prove that the quotient
P GLd (Fq [t])\Bd is weak Ramanujan when d = 3, 4. This is joint work with S. Kwon.
Keywords: Ramanujan complex, Bruhat-Tits building, spectrum of adjacency operators
Kota Saito (University of Tsukuba)
Title: The simple normality of the square root of two and the Riemann zeta function
Abstract: A real number x is called a simply normal number to base b if the base-b expansion ofx has each digit 0, 1,√. . . , b − 1 appearing with average frequency tending to 1/b. We do not knowwhether π, e, log 2, 2, and so on are simply normal or not. In this talk, we report discovering√a relation between the frequency that the digit 1 appears in the binary expansion of 2 anda mean value of the Riemann zeta function on arithmetic progressions. As a consequence, weshow that()1 ∑2nπi (−1)nlimζ
=0l→∞ llog 2nl√0<|n|≤2

if and only if 2 is simply normal to base 2. This is joint work with Yuya Kanado (NagoyaUniversity).
Keywords: simple normality, Riemann zeta function, Perron’s formula, functional equation

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