Hae-Sang Sun

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

We formulate a thermodynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi-Vallée. We introduce the modular partitions of continued fractions and observe that the statistics for modular symbols follow from the behavior of modular partitions. We prove the limit Gaussian distribution and residual equidistribution for modular partitions as a vector-valued random variable on the set of rationals whose denominators are up to a fixed positive integer by st

Let $p$ be an odd prime. We show that the compositum of the Hecke field of a normalized Hecke eigen cuspform for ${\\rm GL}(2)$ over $\\Bbb{Q}$ and a cyclotomic field of a $p$-power degree over $\\Bbb{Q}$, namely the cyclotomic Hecke field, is generated by a single algebraic critical value of the corresponding $L$-function twisted by a Dirichlet character of sufficiently large $p$-power conductor when the level of cuspform is relatively prime to $p$. The same result holds when the level is divis

Abstract. For two different prime numbers p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℓ</m:mi> </m:math> $\ell $ , the special values of Dirichlet L -functions in a finite field of characteristic p are considered as a function on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℓ</m:mi> </m:math> $\ell $ -power roots of unity, which is an analogue of the Iwasawa power series. In this setting, we prove various properties of these functions, for example, transc

We present a concise proof of the conjecture of Mazur-Rubin-Stein on the distribution of modular symbols.

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

We record two remarks on the work of Baladi–Vallée [J.&nbsp;Number Theory 110 (2005), 331–386]. They proved the asymptotic Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers whose denominators ar

We deduce the transcendence of the Iwasawa power series from Borel’s conjecture, namely, the normality of the irrational algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic integers.

"A Group Whose Squares Generate a Dicyclic Group." The American Mathematical Monthly, 78(9), pp. 992–993