arXiv subjects

Ying Wai Lee

Publications and source records attributed to Ying Wai Lee.

10 recordsLinked to original sources

arXiv2026

Exact universal normalizations for the Gál--Koksma lemma

The Gál--Koksma lemma is a standard tool for converting quadratic-mean estimates on consecutive blocks into almost-everywhere bounds for partial sums, without assumptions of independence, mixing, or orthogonality. A natural open problem is to determine exactly which universal growth normalizations are forced by this hypothesis alone. The corresponding universal normalization problem under the abstract consecutive-block second-moment hypothesis is resolved by characterizing exactly which non-decreasing normalizations are valid uniformly over the entire admissible class. The resulting necessary-and-sufficient summability criterion is sharp even for bounded exactly centred systems with constant majorants and exact linear block variance, and determines the critical logarithmic and iterated-logarithmic thresholds.

math.NT↗

arXiv2026

Prescribed realisation of longest runs in continued fractions

Exceptional longest-run behaviour in continued fraction expansions is studied through the interaction between fixed-symbol runs and the overall longest run. For every prescribed partial quotient value and every admissible growth scale, a full Hausdorff dimensional set of irrational numbers is constructed on which the longest run of the prescribed value has exactly the prescribed asymptotic growth and, for every initial length, uniquely realises the overall maximum. It follows that the symbol responsible for the overall longest run can be fixed in advance without any loss of Hausdorff dimension. Thus, the known full-dimensional exceptional-set results for fixed-symbol longest-run growth and for overall longest-run growth are simultaneously strengthened, while the maximising symbol in the overall problem is shown to be fully prescribable.

math.NT↗

arXiv2026

Decomposition of real numbers into sums of Lüroth sets

We study the decomposition of real numbers into sums of Lüroth sets, which are defined by numbers whose Lüroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of Lüroth sets, including summands with digits bounded above, below, and combinations of the two. We also analyse the Hausdorff dimension of Lüroth sets and their sums. The results extend classical findings on continued fractions to Lüroth expansions.

math.NT↗

arXiv2026

Prescribed distinct-digit growth in countable alphabets

The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.

math.DS↗

arXiv2026

Quantitative longest-run laws for partial quotients

Two longest-run statistics are studied: the longest run of a fixed value and the longest run over all values. Under quantitative mixing and exponential cylinder estimates for constant words, a general theorem is proved. Quantitative almost-sure logarithmic growth is obtained, and eventual two-sided bounds with double-logarithmic error terms are established. For continued-fraction partial quotients, explicit centring constants and double-logarithmic error bounds are derived for both statistics.

math.NT↗

arXiv2025

Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures

This paper focuses on the metric properties of Lüroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarník--Besicovitch Theorem, and the result of Dodson. A supplementary proof is provided for a measure-theoretic statement originally proposed by Tan--Zhou. The Beresnevich--Velani Mass Transference Principle is applied to extend a dimensional result of Cao--Wu--Zhang. A counterexample is constructed, leading to a revision of a conjecture by Tan--Zhou concerning dimension, along with a partial result.

math.NT↗

arXiv2024

Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on self-similar probability measures defined on self-similar sets. Explicit upper bounds are derived for their decay rates, improving upon prior research. These findings are illustrated with an application to sets of numbers whose digits in their Lüroth representations are restricted to a finite set.

math.CA↗

arXiv2024

Effective Results in The Metric Theory of Quantitative Diophantine Approximation

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results does not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. This paper addresses this problem and makes the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most $δ>0$, an arbitrarily small constant chosen in advance. We deduce from this the fully effective results for Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative results on $M_{0}$-sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers.

math.NT↗

arXiv2021

Empirically Improved Tokuda Gap Sequence in Shellsort

Experiments are conducted to improve Tokuda (1992) gap sequence in Shellsort into $γ$-sequences, and the best result is the gap sequence in which the $k$-th increment $h_k$ is given by \begin{align} h_k=\left\lceil \frac{γ^k-1}{γ-1} \right\rceil \end{align} , where $γ=2.243609061420001...$ and $k\in\mathbb{N}_1$. The first few increments of the gap sequence are \begin{align} 1,\, 4,\, 9,\, 20,\, 45,\, 102,\, 230,\, 516,\,1158,\,2599,\,5831,\,13082,\,29351,\,65853,\, 147748,\,331490,\,743735,\, ...\end{align}It empirically yields less numbers of comparison on average than Tokuda (1992) gap sequence. In the procedure of search, it reveals the potential existence of a new type of fractal.

cs.DS↗

arXiv2021

Optimal Gap Sequences in Shellsort for $n\leq16$ Elements

Optimal gap sequences in Shellsort, defined as gap sequences having the minimised maximum number of comparisons for a fixed number of pairwise distinct elements, are found by minimax search in reduced permutational spaces, namely Bad $(s,1)$-sorted permutations. Exact optimal gap sequences in Shellsort for $n\leq16$ pairwise distinct elements are established, and the best known gap sequences for $17\leq n\leq 30$ are listed with conjectures made. It notably discovers some optimal gap sequences consist of increments larger than the half of the total number of the elements to sort.

math.CO↗