Diophantine Analysis and Related Fields 2019
Abstracts of the talks

Thursday 7th March
10:00–10:50 Makoto Kawashima (Osaka University)
TITLE: Linear independence measures for values of power of logarithm functions
Abstract: In this talk, we construct (type I) Padé approximations of power oflogarithm functions using that of exponential functions. Applying these approximations, we give new linear independence measures of values of logarithm
functions at algebraic numbers with suﬃciently small absolute value in bothcomplex and p-adic cases. If time permits, we also discuss q-analogue of theseresults.
11:00-11:50 Alan Filipin (University of Zagreb)
TITLE: A polynomial variant of the problem of Diophantus
Abstract: In this talk we prove that if {a, b, c, d} is a set of four non-zero elements of R[X], not all constant, such that the product of any two of its distinct
elements increased by 1 is a square of an element of R[X], then(a + b − c − d)2 = 4(ab + 1)(cd + 1).
Some consequences of the above result are that for an arbitrary positive
integer n there does not exist a set of five non-zero elements from Z[X], which
are not all constant, such that the product of any two of its distinct elements
increased by n is a square of an element of Z[X]. Furthermore, there can exist
such a set of four non-zero elements of Z[X] if and only if n is a square.
Moreover, we prove the same result for the polynomials with coeﬃcientsfrom the ring of Gaussian integers.
This is joint work with Ana Jurasić.
14:00-14:50 Yasutsugu Fujita (Nihon University) (joint work with FlorianLuca)
TITLE: Non-existence of Diophantine quadruples consisting of Fibonacci numbers
Abstract: A set {a1 , . . . , am } of m positive integers is called a Diophantine
m-tuple if ai aj + 1 is a perfect square for all i, j with i < j. The simplest
Diophantine quadruple {1, 3, 8, 120} was found by Fermat. Baker and Davenport proved in 1969 that the extension of the triple {1, 3, 8} to a Diophantine
quadruple must be {1, 3, 8, 120}. Various kinds of generalizations of this result exist. One of them states that if {F2n , F2n+2 , F2n+4 , d} is a Diophantine
1quadruple with n a positive integer and Fk the kth Fibonacci number, thend = 4F2n+1 F2n+2 F2n+3 , which was shown by Dujella in 1999. In this direction,He, Luca and Togbé showed in 2016 that if {F2n , F2n+2 , Fk } is a Diophantine
triple with n > 1, then k = 2n − 2 or 2n + 4. They also conjectured that there
are no Diophantine quadruples consisting of Fibonacci numbers. In 2017, weshowed that there are only finitely many Diophantine quadruples consisting ofFibonacci numbers. In this talk, we give a proof of the conjecture above. The
key to the proof is to apply Baker’s method on linear forms in logarithms repeatedly to an equation satisfied by a binary recurrence sequence and the Fibonaccisequence.
15:00-15:50 Thomas Stoll (Université de Lorraine)
TITLE: The sum of digits in two diﬀerent bases
Abstract: Let sa (n) denote the sum of digits of an integer n in the base a expansion. We show that, provided a and b are multiplicatively independent integers,
any positive real number is a limit point of the sequence {sb (n)/sa (n)}n . We also
provide upper and lower bounds for the counting functions of the corresponding
subsequences. The proofs involve exponential sums, discrepancy estimates aswell as the irrationality measure of log a/ log b. This is joint work with R. de laBretèche and G. Tenenbaum.
16:00-16:50 Ryuji Abe (Tokyo Polytechnic University)
TITLE: Markoﬀ spectrum and arithmetic Fricke groups
Abstract: The Markoﬀ spectrum is defined as the set of the normalized values of arithmetic minima of indefinite quadratic forms. Fricke groups are twogenerator free Fuchsian groups the quotient spaces of which are once punctured
tori. It is well-known that the discrete part of the Markoﬀ spectrum is interpreted in terms of simple closed geodesics on the once punctured torus of anarithmetic Fricke group which is a subgroup of the modular group with index 6.
In this talk, we report an interpretation of sequences in not discrete part of theMarkoﬀ spectrum using other arithmetic Fricke groups. We use a graph representation of Diophantine equations whose integer solutions give the sequencesin the Markoﬀ spectrum. This talk is based on joint work with I. R. Aitchison
and B. Rittaud.
17:00-17:50 Iekata Shiokawa (Keio University) (joint work with Daniel Duverney and Takeshi Kurosawa)
TITLE: Irrationality exponents of certain sums of rational numbers
Abstract: Let {xn }n≥0 be a sequence of rational numbers greater than one suchthat zn+1 = xn+1 x−2n ≥ 1 for all suﬃciently large n and let εn ∈ {−1, 1} with
ε1 = 1. Define δ1 = den x1 and δn+1 = δn2 den zn+1 (n ≥ 1), where den(a/b) = |b|
if a, b ∈ Z. Assume∑∞ that log δn+1 = o(log xn ). Then the irrationality exponentof the number n=1 εn /xn is equal to lim supn→∞ log xn+1 / log xn , where theirrationality exponent of irrational number α is defined by the supremum of theset of numbers µ for which the inequality |α − p/q| < q −µ has infinitely many
rational solutions p/q with q > 0.

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Friday 8th March
9:00-9:50 Yohei Tachiya (Hirosaki University)
TITLE: Refinement of Chowla-Erdős method and arithmetical properties of certain Lambert series
Abstract: In this talk, we refine the method of Chowla and Erdős on the irrationality of Lambert series. As applications of our main theorem, we give linear
independence results for various kinds of Lambert series. Moreover, we show
the irrationality of certain reciprocal sums of Fibonacci and Lucas numbers associated with prime numbers. This is a joint work with Daniel Duverney andYuta Suzuki.
10:00-10:50 Dong Han Kim (Dongguk University)
TITLE: On the set of inhomogeneous Diophantine approximations
Abstract: Fix an irrational number θ. We consider the set of points y such that∥nθ − y∥ < φ(n) with monotone error functions φ(n). We give a necessary andsuﬃcient condition for the set has full Lebesgue measure and calculate the Hausdorﬀ dimension. We also consider the dimension of the uniform inhomogeneousapproximation and the dimension of badly approximable sets, which depend on
the Diophantine property of the irrational number θ. This talk is based on joint
works with Yann Bugeaud, Michael Fuchs, Lingmin Liao, Seonhee Lim, MichalRams and Baowei Wang.
11:00-11:50 Teturo Kamae (Osaka City University)
TITLE: Normal numbers as configurations on N2
Abstract: Let A = {0, 1, · · · , d−1} be an alphabet. We consider a configuration
2x ∈ AN , where N = {1, 2, · · · }. We denote its restriction to a rectangle C ×D ⊂N2 by x[C × D] ∈ AC×D . We denote its translation by (i, j) ∈ (N ∪ {0})2 asT (i,j) x[C × D] ∈ A(C+i)×(D+j) . That is,T (i,j) x[C × D](u + i, v + j) = x[C × D](u, v)
for any (u, v) ∈ C × D. Let∪A+ =A[n]×[m] ,n=1,2,··· ;m=1,2,···where we denote [n] = {1, 2, · · · , n}. For η ∈ AC×D and ξ ∈ A[n]×[m] , we saythat ξ is a (i, j)-prefix of η denoting ξ ≺(i,j) η if ([n] + i) × ([m] + j) ⊂ C × D andη[([n] + i) × ([m] + j)] = T (i,j) ξ. We denote by |η|ξ the number of occurrencesof ξ in η. That is,|η|ξ = #{(i, j); ξ ≺(i,j) η}.
We call x ∈ AN normal if2lim1N,M →∞ N M|x[[N ] × [M ]]|ξ = d−nm

holds for any n, m ∈ N and ξ ∈ A[n]×[m] . Let X be a AN -valued randomvariable such that {X(n, m); (n, m) ∈ N2 } are i.i.d. random variables with2
P (X(0, 0) = a) = 1/d for any a ∈ A. Then, almost all realizations x ∈ AN ofX are normal.
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We consider a stronger notion than normality. For x ∈ AN , we define∑ΣN M (x) =|x[[N ] × [M ]]|2ξ .
2ξ∈A+Then, almost all realizations x ∈ AN of X satisfy that2∞

11 ∑ 1
NMlim+.
Σ(x)=N,M →∞ N 2 M 2
4dk − 1k=1

We call x ∈ AN as this Σ-random. Then, Σ-random implies normal, but theconverse is not true. We study the normality and the Σ-randomness in thissetting.
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14:00-14:50 Alan Haynes (University of Houston)
TITLE: Diophantine approximation and diﬀraction from quasicrystals
Abstract: In this talk we will explain how diﬀraction patterns observed from cutand project sets (models for physical materials called quasicrystals) are determined by Diophantine approximation properties of the underlying constructions.
The classical approach to calculating diﬀraction patterns seen from these objects
assumes an infinite model, and for this reason it is not the most practical, froman experimental point of view. Our approach (joint work with Michael Baake)quantifies precisely how much the diﬀraction patterns observed from finite patterns in cut and project sets deviate from the infinite models. Our methods areexplicit and geared towards numerical computation, and they demonstrate theimportance of Diophantine approximation to accurately determining complex
phases and amplitudes of these diﬀraction images.
15:00-15:50 Hohto Bekki (The University of Tokyo)
TITLE: On the geodesic continued fraction and its periodicity
Abstract: The classical Lagrange theorem in the theory of continued fraction
says that (1) the continued fraction expansion of a given real number becomes
periodic if and only if the number is a real quadratic irrational, and (2) theperiod of the continued fraction expansion actually gives the fundamental unitof the associated (order of) real quadratic field. On the other hand, we know that
(1) a geodesic on the upper-half plane becomes a closed geodesic (“periodic”)
on the modular curve if and only if the two endpoints are the conjugate real
quadratic irrationals, and (2) the length of the closed geodesic becomes theregulator of the associated (order of) real quadratic field. Based on this analogy,
we have studied generalizations of continued fraction using the geodesics on somelocally symmetric spaces, and established the generalizations of the Lagrange
theorem. In this talk we mainly discuss the case where the locally symmetricspace is the Shimura curve coming from the (2,3,7)-triangle group.
16:00-16:50 Arnaldo Nogueira (Institut de Mathématiques de Marseille)
TITLE: On the action of the semigroup of non singular integral matrices on Rn
Abstract: Let Γ be the multiplicative semigroup of all n × n matrices with integral entries and nonzero determinant. Let 1 ≤ p ≤ n − 1 and V = Rnp =4

Rn ⊕ · · · ⊕ Rn (p copies). Consider the action of Γ on V , given by the natural action on each component, by matrix multiplication on the left. Then for
x = (x1 , . . . , x∑p ) ∈ V , the Γ-orbit is dense in V if and only if there is no linear
pcombination j=1 λj xj , with λj ̸= 0 for some j, which is a rational vector innR ; in fact the assertion holds also for the orbit of the group SL(n, Z) that iscontained in Γ. When the Γ-orbit of x is dense, given y ∈ Rnp , and ϵ > 0
one may ask for γ ∈ Γ such that ||γx − y|| < ϵ, with a bound on ||γ|| in termsof ϵ. There has been considerable interest in the literature in quantitative results of this kind, for various group actions. In particular it was shown by
Laurent-Nogueira, for n = 2, that given an irrational vector x in R2 , any target
vector y ∈ R2 and ρ < 13 there exist infinitely many γ in SL(2, Z) such that||γx − y|| ≤ ||γ||−ρ . In the talk we will describe some results along this theme forthe action of Γ; for the case n = 2 the result is stronger in import than what isrecalled above for SL(2, Z), in the sense that the corresponding statement holds
for all ρ less than 1, in place of 13 for SL(2, Z).
The talk is based on a joint work with S.G. Dani.
17:00-17:50 Haruki Ide (Keio University)
TITLE: Algebraic independence of the values and the partial derivatives of a certain entire function of two variables
Abstract: In previous works, Tanaka constructed a certain entire function withthe property that its values and its derivatives of any order at any distinct
nonzero algebraic numbers are algebraically independent. He also constructedan entire function defined by an infinite product and having the property thatits values and its derivatives of any order at any distinct nonzero algebraic numbers except zeroes of the infinite product are algebraically independent. In thistalk, the speaker will introduce a certain entire function of two variables withthe following remarkable properties:(1) All the numbers whose algebraic independency was proved in Tanaka sprevious works stated above are given as the values or the partial derivatives ofthis two-variable function at algebraic points.
(2) The infinite set consisting of the values and the partial derivatives of any order of this two-variable function at any distinct algebraic points is algebraicallyindependent.

Saturday 9th March
10:00-10:50 Tomohiro Ooto (Tecnos Data Science Engineering)
TITLE: Diophantine exponents for hyperquadratic continued fractions
Abstract: Hyperquadratic continued fractions are some class of algebraic continued fractions in the field of Laurent series over a finite field. Many explicit
continued fractions are known for nonquadratic but hyperquadratic elements.
In this talk, we estimate Diophantine exponents wn and wn∗ for such hyperquadratic continued fractions. As its applications, we determine the degrees ofsuch hyperquadratic continued fractions. This is a joint work with K. Ayadi(Univ. of Sfax).

511:00-11:50 Takafumi Miyazaki (Gunma University)
TITLE: Application of cubic residue theory to a special type of unit
equation concerning Eisenstein triples
Abstract: A special type of unit equation is of the form a + b = c, where a, b, c
are pair-wise coprime positive integer variables satisfying the condition that any
prime dividing abc belongs to a given finite set S of primes. As an application tothe celebrated theorem of Thue-Siegel-Roth, it was shown that the mentioned
unit equation has only finitely many solutions (a, b, c), and the number of such
triples is at most an explicit constant depending only on S. Though the theoryof linear forms in complex logarithms of Baker can be applied to estimate thesize of possible triples (a, b, c), it is still not easy to describe all such triples evenfor very special cases.
In this talk, as a very special case of the mentioned unit equation, we treat
the Diophantine equation Ax + B y = C z , where A, B, C are given integers
with min{A, B, C} > 1 and gcd(A, B, C) = 1. We solve this equation for somefamily of triples (A, B, C) such that A, B, C forms three length of a triangle
having a 2π/3 angle. A key of the proof is to compute values of cubic residue
characters. The result can be regarded as a relevant analogue to some resultson Jeśmanowicz’ conjecture concerning primitive Pythagorean triples.
12:00-12:50 Hajime Kaneko (University of Tsukuba)TITLE: Analogy of the Lagrange spectrum for powers of quadratic
Pisot units
Abstract: For a real number x, let ∥x∥ be the distance of x and its nearest
integer. The Lagrange spectrum is related to the Diophantine approximation
property of the arithmetic sequences (nα)n=0,1,... . More precisely, for any irrational number α, let L(α) = 1/cα , where cα = lim inf n→∞ n∥nα∥. Then theLagrange spectrum is defined by L = {L(α) | α ∈ R\Q}.
In this talk we consider analogy of the Lagrange spectrum for geometricsequences. Little is known about Diophantine approximation properties ofgeometric sequences. For instance, it is still open problem to disprove thatlimn→∞ ∥en ∥ = 0. The main purpose of this talk is to investigate the set{lim supn→∞ ∥ξαn ∥ | ξ ∈ R}, where α is a quadratic Pisot unit.
This is a joint work with Shigeki Akiyama and Teturo Kamae.

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