Functional spaces and their exceptional sets

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)].

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Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Open Mathematics ◽

10.1515/math-2017-0130 ◽

2017 ◽

Vol 15 (1) ◽

pp. 1517-1529

Author(s):

Zhao Feng

Keyword(s):

Short Intervals ◽

Exceptional Sets ◽

Goldbach Problem ◽

Almost All

Abstract In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

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Exceptional sets related to the largest digits in Lüroth expansions

International Journal of Number Theory ◽

10.1142/s1793042122500737 ◽

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.

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