(Japanese / 日本語)
(Japanese / 日本語)
Speaker: Kota Saito (Nagoya University)
Date: 2020 Dec. 12th (Sat) from 16:00 JST
Title: Linear Diophantine equations in Piatetski-Shapiro sequences
Abstract: A Piatetski-Shapiro sequence with exponent α is a sequence of integer parts of n^α (n=1,2,…) with a non-integral α>0. We let PS(α) denote the set of those terms. In this talk, we study the set of α so that the equation ax+by=cz has infinitely many pairwise distinct solutions (x,y,z)∈PS(α)^3, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many α>2 such that PS(α) contains infinitely many arithmetic progressions of length 3. This is a joint work with Toshiki Matsusaka (Nagoya University).
Speaker: Yu Yasufuku (Nihon University)
Date: 2021 Jan. 23rd (Sat) from 16:00 JST
Title: Expanding Corvaja--Zannier's S-unit GCD Inequality
Abstract: Corvaja--Zannier proved that GCD(u-1, v-1) is small compared to the heights of u and v when u and v are S-units. In this talk, we analyze the same quantity when u and v are not assumed S-units. The obtained inequality is weaker than what is conjectured by Vojta, but in some sense stronger than what is obtained earlier by Luca. Just like the Corvaja--Zannier proof, the main ingredient is the Schmidt subspace theorem, but we use it through the machinery developed recently by Ru--Vojta.
Speaker: Anthony POELS,
JSPS researcher in Nihon University, University of Ottawa (Canada), ENS Lyon (France)
Date: 2021 Feb. 20th (Sat) from 16:00 JST
Title: On Sturmian type numbers
Abstract: In 1969, Davenport and Schmidt gave a non-trivial upper bound > 1/2 for the uniform exponent of simultaneous rational approximation to a given transcendental real number and to its square. For a long time it was conjectured that the aforementioned exponent was always equal to 1/2, the lower bound given by Dirichlet's approximation theorem. However, in 2004 Roy proved that this conjecture was false by constructing real numbers - called extremal numbers - whose uniform exponent is precisely equal to Davenport and Schmidt's upper bound. In this talk, we will first present the ideas behind Davenport and Schmidt's inequality and Roy's extremal numbers. Then, we will generalize Roy's construction to build a larger family of numbers called Sturmian type numbers.
Speaker:Yusuke Tanuma (Keio University)
Date: 2021 March 13th (Sat) from 16:00 JST
Title: Arithmetic properties of the values of Hecke-Mahler series for several quadratic irrational numbers
Abstract: Hecke-Mahler series is the generating function of the sequence {[kω]} of integral parts of positive integral multiples of a real number ω. The arithmetic properties of its values have been studied by several authors. Adamczewski and Faverjon treated the algebraic independence of the values of Hecke-Mahler series for several quadratic irrational numbers generating different quadratic fields. In this talk, we study the algebraic independence of the values of Hecke-Mahler series for several quadratic irrational numbers generating the same quadratic field.