過去のDarfセミナーの記録を記載します。
第一回目
講演者: 齋藤耕太 (名古屋大学)
日時: 2020年12月12日(土)16:00から
タイトル: Linear Diophantine equations in Piatetski-Shapiro sequences
アブストラクト:
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).
第二回目
講演者: 安福 悠 (日本大学)
Yu Yasufuku (Nihon University)
日時: 2021年1月23日(土)16:00から
タイトル: Expanding Corvaja--Zannier's S-unit GCD Inequality
アブストラクト:
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.
第三回目
講演者: Anthony POELS,
JSPS researcher in Nihon University,
University of Ottawa (Canada),
ENS Lyon (France)
日時: 2月20日(土)16:00から
タイトル: On Sturmian type numbers
アブストラクト: 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.
第四回目
講演者: 田沼優佑 (慶應義塾大学)
Yusuke Tanuma (Keio University)
日時: 3月13日(土)16:00から
タイトル: Arithmetic properties of the values of Hecke-Mahler series
for several quadratic irrational numbers
アブストラクト:
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.
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