A NOTE ON A COMPLETE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2018 ◽  
Vol 98 (1) ◽  
pp. 60-63
Author(s):  
DIEGO MARQUES ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Recent Result ◽  
Real Number ◽  
Complete Solution ◽  
Short Note ◽  
Radius Of Convergence ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Functions

Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$. This solves the question posed by K. Mahler completely.

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

Exceptional sets related to the largest digits in Lüroth expansions

2021 ◽  
pp. 1-15
Author(s):  
Shuyi Lin ◽  
Jinjun Li ◽  
Manli Lou
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Disjoint Union ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Lüroth Expansion

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

scholarly journals NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

2010 ◽  
Vol 88 (3) ◽  
pp. 353-361
Author(s):  
R. G. HALBURD ◽  
R. J. KORHONEN
Keyword(s):  
Maximum Modulus ◽  
Point Of View ◽  
Modulus Function ◽  
Linear Measure ◽  
Real Analysis ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Finite Linear ◽  
Increasing Function ◽  
Nondecreasing Functions

AbstractAccording to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals ALGEBRAIC VALUES OF TRANSCENDENTAL FUNCTIONS AT ALGEBRAIC POINTS

2010 ◽  
Vol 82 (2) ◽  
pp. 322-327 ◽  
Author(s):  
JINGJING HUANG ◽  
DIEGO MARQUES ◽  
MARTIN MEREB
Keyword(s):  
Entire Function ◽  
Transcendental Entire Function ◽  
General Version ◽  
Exceptional Set ◽  
Transcendental Functions

AbstractIt is shown that any subset of $\overline {\mathbb {Q}}$ can be the exceptional set of some transcendental entire function. Furthermore, we give a much more general version of this theorem and present a unified proof.

scholarly journals THE EXCEPTIONAL SETS ON THE RUN-LENGTH FUNCTION OF β-EXPANSIONS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750060 ◽  
Author(s):  
LIXUAN ZHENG ◽  
MIN WU ◽  
BING LI
Keyword(s):  
Hausdorff Dimension ◽  
Maximal Length ◽  
Length Function ◽  
Exceptional Set ◽  
Run Length ◽  
Exceptional Sets ◽  
Monotonically Increasing Function ◽  
Increasing Function

Let [Formula: see text] and the run-length function [Formula: see text] be the maximal length of consecutive zeros amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. The exceptional set [Formula: see text] is investigated, where [Formula: see text] is a monotonically increasing function with [Formula: see text]. We prove that the set [Formula: see text] is either empty or of full Hausdorff dimension and residual in [Formula: see text] according to the increasing rate of [Formula: see text].

scholarly journals A short note on hit-and-miss hyperspaces

2003 ◽  
Vol 4 (2) ◽  
pp. 281
Author(s):  
René Bartsch ◽  
Harry Poppe
Keyword(s):  
Recent Result ◽  
Short Note

<p>Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “κ-Lindelöfness” and “κ-compactness” for cardinals κ. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of Som Naimpally, to who this article is hearty dedicated.</p>

scholarly journals Note on Long Paths in Eulerian Digraphs

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Charlotte Knierim ◽  
Maxime Larcher ◽  
Anders Martinsson
Keyword(s):  
Recent Result ◽  
Short Note ◽  
Average Degree ◽  
Logarithmic Factor ◽  
Long Paths

Long paths and cycles in Eulerian digraphs have received a lot of attention recently. In this short note, we show how to use methods from [Knierim, Larcher, Martinsson, Noever, JCTB 148:125--148] to find paths of length $d/(\log d+1)$ in Eulerian digraphs with average degree $d$, improving  the recent result of $\Omega(d^{1/2+1/40})$. Our result is optimal up to at most a logarithmic factor.  

Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

2001 ◽  
Vol 44 (2) ◽  
pp. 150-159 ◽  
Author(s):  
Piotr Jakóbczak
Keyword(s):  
Unit Ball ◽  
Lebesgue Measure ◽  
Bergman Space ◽  
Dimensional Subspace ◽  
Natural Topology ◽  
Exceptional Set ◽  
One Dimensional ◽  
Exceptional Sets

AbstractLet BN be the unit ball in and let f be a function holomorphic and L2-integrable in BN. Denote by E(BN, f) the set of all slices of the form , where L is a complex one-dimensional subspace of , for which is not L2-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for f. We give a characterization of exceptional sets which are closed in the natural topology of slices.

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