nonnegative number

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000173 ◽

2021 ◽

pp. 1-29

Author(s):

LINGLING HUANG ◽

CHAO MA

Keyword(s):

Growth Rate ◽

Hausdorff Dimension ◽

Natural Number ◽

Continued Fractions ◽

Continued Fraction ◽

Irrational Number ◽

Continued Fraction Expansion ◽

Nonnegative Number ◽

Partial Quotients

Abstract This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

Combinatorics of poly-Bernoulli numbers

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2015.52.4.1325 ◽

2015 ◽

Vol 52 (4) ◽

pp. 537-558 ◽

Cited By ~ 1

Author(s):

Beáta Bényi ◽

Péter Hajnal

Keyword(s):

Recursive Formula ◽

Natural Generalization ◽

Bernoulli Numbers ◽

Combinatorial Interpretation ◽

Nonnegative Number ◽

New Interpretation

The Bn(k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (Bn = Bn(1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn(k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn(−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

Computer Science and Communications Dictionary ◽

10.1007/1-4020-0613-6_12443 ◽

2000 ◽

pp. 1108-1108

Author(s):

Martin H. Weik

Keyword(s):

Nonnegative Number

ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000497 ◽

2018 ◽

Vol 98 (2) ◽

pp. 212-220

Author(s):

ARTŪRAS DUBICKAS

Keyword(s):

Positive Integer ◽

Asymptotic Formula ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Algebraic Numbers ◽

Bounded Degree ◽

Nonnegative Number ◽

Weil Height

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].