Diophantine Analysis and Related Fields 2023
Room 11, 2nd Floor, Faculty of Science and Technology Bldg. No. 2,Bunkyo-cho Campus, Hirosaki University,Hirosaki, Japan

Abstracts of the Talks
Friday 3rd March
14:30–15:20 István Pink (Univ. Debrecen)
Title: Number of solutions to a special type of unit equations in two unknowns II
Abstract: This talk is a continuation of the one given by myself on April 17, 2021 at the online
Darf seminar. The topic is the best possible general estimate on the number of solutions toa special type of unit equations in two unknowns over the rationals. R. Scott and R. Styer
conjectured in 2016 that for any fixed relatively prime positive integers a, b and c greater than
1 the equation ax + by = cz has at most one solution in positive integers x, y and z, except for
specific cases. In this talk we give a brief introduction to the conjecture, and present our resultswith their proofs, which in particular provides an analytic proof of the celebrated theorem of
Scott (1993) solving the conjecture for c = 2 in a purely algebraic manner. This is a joint workwith Takafumi Miyazaki (Gunma University).
15:40-16:30 Koichi Kawada (Iwate Univ.)
Title: An overview of the Waring-Goldbach problem
Abstract: The talk begins with an overview of the current state of research on representationsof numbers by sums of powers of primes, for general larger exponents. Then I turn to the casesof smaller exponents, especially sums of cubes of primes, and talk on the result obtained jointlywith Lilu Zhao, that every sufficiently large even number can be expressed as x3 + p31 + p32 +· · · + p37 with primes pi and a P2 -number x.
16:50-17:30 Yusuke Washio (Nihon Univ.)
Title: A linear independence measure of polylogarithms relying on Padé approximation
Abstract: We obtain a linear independence measure related to polylogarithms and values ofLerch functions at algebraic points, by using Padé approximation. We also talk about new examples of independence measure concerning with the criterion obtained by S. David (SorbonneUniv.), N. Hirata-Kohno (CST, Nihon Univ.) and M. Kawashima (CIT, Nihon Univ.). This is ajoint work with Ryuji Muroi (CST, Nihon Univ.).

Saturday 4th March
9:30-10:20 Daniel Duverney (Baggio Engineering School)
Title: Sylvester sequence, irrationality and transcendence
Abstract: Let (Sn )n≥0 be the Sylvester sequence, defined by
S0 = 2,

Sn+1 = Sn2 − Sn + 11(n ≥ 0) .

The purpose of this talk is to present some results on the irrationality and transcendence of thenumbers∞∑an
S + bnn=0 n)∞ (∏an, Sn + an + bn ̸= 0 (n ≥ 0)
G=1+Sn + bn
n=0a1
ana0,
H=S0 + b0 + S1 + b1 + · · · + Sn + bn + · · ·F =where an and bn are integers satisfying
Sn + bn ̸= 0 (n ≥ 0)

andlog (max{1, |an | , |bn |}) = o (2n ) .

We can solve completely the problem of the irrationality of F, G and H. We can also provethat the continued fraction H is transcendental by using Roth theorem. For the transcendenceof F and G (for which Mahler’s method seems to be the best tool), we can prove it only invery special cases. This talk is based in part on joint works with Takeshi Kurosawa and Iekata
Shiokawa.
10:40-11:30 Dong Han Kim (Dongguk Univ.)
Title: The Markoff spectrum on the Hecke group
Abstract: We consider the Markoff spectrum on the Hecke group. The Markoff spectrum on H4
is known as the Markoff spectrum of index 2 sublattices by Vulakh and the Markoff spectrumof 2-minimal forms or C-minimal forms by Schmidt, who characterized the spectrum up to thefirst accumulation point. The Markoff spectrum on H6 is also known as the Markoff spectrum of3-minimal forms by Schmidt. After the first accumulation point, we show that both spectra havepositive Hausdorff dimension and find gaps. This talk is based on joint work with Byungchul
Cha and Deokwon Sim.
11:50-12:30 Haruki Ide and Taka-aki Tanaka (Keio Univ.)
Title: Mahler’s method for algebraic independence of partial derivatives of certain series
in several variables
Abstract: Some entire functions are known to have the properties that all the values of successive derivatives of each of those functions at any nonzero algebraic numbers are algebraicallyindependent. Examples of such entire functions were shown independently by Nishioka andthe second speaker in the case of one variable and in the case of two variables by the firstspeaker. In this talk we construct an entire function of arbitrary number of variables having
the following property: The infinite set consisting of all the values of all its partial derivativesof any orders at all algebraic points, including zero components, is algebraically independent.
One of the key points in the proof of this result is to find a biregular transformation between thepartial derivatives of the entire function in question and those of a certain Lambert type series inseveral variables. Using this biregular transformation, we reduce the problem to the algebraic
independency of the values of the partial derivatives of the Lambert type series, which can berepresented as special values of certain Mahler functions. Another key point in the proof is to2

prove the linear independency of the Mahler functions themselves, for which we introduce anew method.
14:30-15:20 Damien Jamet (Univ. Lorraine)
Title: On a probabilistic extension of the Oldenburger-Kolakoski sequence
Abstract: The Oldenburger-Kolakoski sequence is the only infinite sequence over the alphabet{1, 2} that starts with 1 and is its own run-length encoding. In the present work, we take a
step back from this largely known and studied sequence by introducing some randomness inthe choice of the letters written. This enables us to provide some results on the convergenceof the density of 1’s in the resulting sequence. When the choice of the letters is given by
an infinite sequence of i.i.d. random variables or by a Markov chain, the average densities ofletters converge. Moreover, in the case of i.i.d. random variables, we are able to prove that thedensities even almost surely converge.
15:40-16:20 Hiroaki Ito (Univ. Tsukuba)
Title: Symmetry of continued fraction algorithms
Abstract: F. Schweiger introduced a fibred system of multidimensional continued fraction algorithms. An advantage of a fibred system is that it often provides a systematic constructionof absolutely continuous invariant measure by using a dual system. We can find a symmetry
in measure by investigating the dual algorithms, I will talk about the symmetry of several algorithms. Also, I present examples with partial self-dual and examples that are ergodic but not
self-dual.
16:40-17:20 Kota Saito (Univ. Tsukuba)
Title: Finiteness of solutions of Diophantine equations on Piatetski-Shapiro sequences
Abstract: A sequence of positive integers of the form ⌊nα ⌋ for some non-integral α > 1 is calleda Piatetski-Shapiro sequence, for short PS-sequence. Let PS(α) = {⌊nα ⌋ : n = 1, 2, . . .}. In
this talk, we discuss the linear equations (E) x + y = z on PS(α). As a main result, we showthat for almost all α > 3, (E) has at most finitely many solutions (x, y, z) ∈ PS(α). If time
permitted, we give a result for general linear equations y = a1 x1 + · · · + an xn and the Hausdorff
dimension.

Sunday 5th March
9:30-10:20 Pierre Popoli (Univ. Lorraine)
Title: On the binary digits of n and n2
Abstract: 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(n2 ) = k, fora given positive integer k. The remaining cases that could not be treated by theses authors werek = 9, 10, 11, 14 or 15. In this talk, I will present the results of our article on the cases k = 9, 10
and 11 and the difficulties to settle for the two remaining cases k = 14 and 15. A related
problem is to study perfect squares of odd integers with four binary digits. Bennett, Bugeaudand Mignotte (2012) proved that there are only finitely many solutions and conjectured that theset of solutions is composed of 13, 15, 47 and 111. In the same paper, we give an algorithm3to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show
related results for perfect squares of odd integers with five binary digits. This is joint work withAloui, Jamet, Kaneko, Kopecki and Stoll.
10:40-11:30 Makoto Kawashima (Nihon Univ.)
Title: On arithmetic properties of hypergeometric functions
Abstract: The classical hypergeometric functions, for instance Gauss hypergeometric function,
are familiar in number theory and many other areas of applications. In this talk, we shall discuss
the arithmetic properties of these functions and their values. In particular, we will report ourrecent results on the linear independence of values of contiguous hypergeometric functions atdistinct points. This is a joint work with S. David and N. Hirata-Kohno.
11:50-12:30 Fan Wen (Univ. Tsukuba)
Title: Hausdorff dimension on Markoff-Lagrange problem
Abstract: We focus on a set L(α) called Lagrange spectrum of geometric progressions, thedefinition of L(α) is as follow:{}nL(α) = lim sup ||ξα || ξ ∈ R ,n→∞where ||ξαn || is the distance from ξαn to the nearest integer.
Some properties of L(α) have been discussed in the history. And what we want to do isestimating the Hausdorff dimension of set L(α) ∩ [0, t] where t ∈ [0, 12 ). In the case that α ≥ 2is an integer, we give a set F (t) which contains L(α) ∩ [0, t] . And we obtain some propertiesof F (t) and give the upper bounds of dim H (F (t)).

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