arXiv subjects

Min Woong Ahn

Publications and source records attributed to Min Woong Ahn.

10 recordsLinked to original sources

arXiv2026

Baire-category transfer between Engel and Pierce expansions

In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding $n-2$ to the $n$th Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.

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arXiv2026

Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions

Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every $x\in[0,1]$, the upper and lower limits of a normalized difference between $Nx$ and the number of leap years up to the year $N$ determined by the Pierce expansion digits of $x$. In this paper, we show that, for every $x\in[0,1]$, both limits are determined by the lower limit of a normalized logarithm of the product of the first $n$ digits of $x$. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in $[0,1]$ and its intersection with any non-empty open subset of $[0,1]$ has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.

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arXiv2026

A topology for Engel expansions: evaluation and digit coding maps

We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals, and give an application to Baire category results for functions of the digits.

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arXiv2025

Möbius inversion and coprime summation for error-sum functions of continued fractions

We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly equal to $1$. Our proof is number-theoretic in nature and involves Möbius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a supplementary result, we rederive the known upper bound of $3/2$ for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.

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arXiv2025

Convergence exponent of Pierce expansion digit sequences

In this paper, we investigate the convergence exponent of Pierce expansion digit sequences. We explore some basic properties of the convergence exponent as a real-valued function defined on the closed unit interval, as well as those of the level sets of the function. Additionally, we further study subsets of the closed unit interval on which the series of positive $s$th powers of the reciprocals of the Pierce expansion digits diverges.

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arXiv2025

An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension

The digits of the Pierce expansion satisfy the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is 1. We provide an elementary proof of this fact by adapting Jun Wu's method, which was originally used for Engel expansions. Our approach emphasizes the fractal nature of exceptional sets and avoids advanced machinery, thereby relying instead on explicit sequences and constructive techniques. Furthermore, our method opens the possibility of extending similar analyses to other real number representation systems, such as the Engel, Lüroth, and Sylvester expansions, thus paving the way for further explorations in metric number theory and fractal geometry.

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arXiv2025

Continuity of the continued fraction mapping revisited

The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction mapping is a homeomorphism onto the product space $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}$ is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval.

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arXiv2025

Exceptional sets to Shallit's law of leap years in Pierce expansions

In his 1994 work, Shallit introduced a rule for determining leap years that generalizes both the historically used Julian calendar and the contemporary Gregorian calendar. This rule depends on a so-called intercalation sequence. According to what we term Shallit's law of leap years, almost every point of the interval $[0,1]$ with respect to Lebesgue measure has the same limsup and liminf, respectively, of a quotient defined in terms of the number of leap years determined by the rule using the Pierce expansion digit sequence as an intercalation sequence. In this paper, we show that the set of exceptions to this law is dense and has full Hausdorff dimension in $[0,1]$, and that the exceptional set intersected with any non-empty open subset of $[0,1]$ has full Hausdorff dimension in $[0,1]$. As a more general result, we establish that for certain subsets of $[0,1]$ concerning the limiting behavior of Pierce expansion digits, intersecting with a non-empty open subset of $[0,1]$ preserves the Hausdorff dimension.

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arXiv2024

Hausdorff dimensions in Pierce expansions

The digits of Pierce expansion obey the law of large numbers, the central limit theorem, and the law of the iterated logarithm in the Lebesgue measure sense. We calculate the Hausdorff dimensions of the exceptional sets of each of the three laws. We further determine the Hausdorff dimensions of certain sets arising in Pierce expansions.

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arXiv2024

On the error-sum function of Pierce expansions

We introduce the error-sum function of Pierce expansions. Some basic properties of the error-sum function are analyzed. We also examine the fractal property of the graph of it by calculating the Hausdorff dimension, the box-counting dimension, and the covering dimension of the graph.

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