arXiv subjects
Publications and source records attributed to Tsz Ho Chan.
In this article, we derive better results concerning powered numbers in short intervals, both unconditionally and conditionally on the $abc$-conjecture. We make use of sieve method, a polynomial identity, and a recent breakthrough result on density of sets with no $k$-term arithmetic progression. In the process, we study integers over short intervals that have with a big smooth divisor.
In this paper, we study numbers $n$ that can be factored in four different ways as $n = A B = (A + a_1) (B - b_1) = (A + a_2) (B - b_2) = (A + a_3) (B - b_3)$ with $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$. We obtain the optimal upper bound $A \le 0.04742 \ldots \cdot C^3 + O(C)$. The key idea is to transform the original question into generalized Pell equations $a x^2 - b y^2 = c$ and study their solutions.
In this paper, we continue the study of three close factorizations of an integer and correct a mistake of a previous result. This turns out to be related to lattice points close to the center point $(\sqrt{N}, \sqrt{N})$ of the hyperbola $x y = N$. We establish optimal lower bounds for $L^1$-distance between these lattice points and the center. We also give some good examples based on polynomials and Pell equations more systematically.
In this paper, we continue the study of small squares containing at least two points on a modular hyperbola $x y \equiv c \pmod{p}$. We deduce a lower bound for its side length. We also investigate what happens if the ``distances" between two such points are special type of numbers like prime numbers, squarefree numbers or smooth numbers as well as more general multiplicatively closed sets or almost dense sets.
This note concerns the non-existence of three consecutive powerful numbers. We use Pell equations, elliptic curves, and second-order recurrences to show that there are no such triplets with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power.
In this paper, we study how short an interval $[x, x + x^θ]$ contains an integer of the form $n_1 n_2 n_3$ and $m_1 m_2 m_3 m_4$ with $n_1 \approx n_2 \approx n_3$ and $m_1 \approx m_2 \approx m_3 \approx m_4$. The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals.
In this article, we are interested in whether a product of three consecutive integers $a (a+1) (a+2)$ divides another such product $b (b+1) (b+2)$. If this happens, we prove that there is some gaps between them, namely $b \gg \frac{a \log a)^{1/6}}{\log \log a)^{1/3}}$. We also consider other polynomial sequences such as $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$ for some fixed integer $l$. Our method is based on effective Liouville-Baker-Feldman theorem.
In this paper, we show that, for some constant $C > 0$, the interval $(x, x + C x^{5/26}]$ always contains a squarefree number when $x$ is sufficiently large (in terms of $C$). Our improvement comes from establishing asymptotic relations between the shifts $a$ and $b$ when $m n^2 \approx (m - a) (n + b)^2$ We apply them to study quadruples $(m + a_1) (n - b_1)^2 \approx m n^2 \approx (m - a_2)(n + b_2)^2 \approx (m - a_2 - a_3)(n + b_2 + b_3)^2$ and generalize Roth differencing and Filaseta-Trifonov differencing by allowing $b_1$ to be different from $b_3$. We also introduce a new differencing and exploit the interplay among these three differencings.
In the domain of chip design, Hardware Description Languages (HDLs) play a pivotal role. However, due to the complex syntax of HDLs and the limited availability of online resources, debugging HDL codes remains a difficult and time-intensive task, even for seasoned engineers. Consequently, there is a pressing need to develop automated HDL code debugging models, which can alleviate the burden on hardware engineers. Despite the strong capabilities of Large Language Models (LLMs) in generating, completing, and debugging software code, their utilization in the specialized field of HDL debugging has been limited and, to date, has not yielded satisfactory results. In this paper, we propose an LLM-assisted HDL debugging framework, namely HDLdebugger, which consists of HDL debugging data generation via a reverse engineering approach, a search engine for retrieval-augmented generation, and a retrieval-augmented LLM fine-tuning approach. Through the integration of these components, HDLdebugger can automate and streamline HDL debugging for chip design. Our comprehensive experiments, conducted on an HDL code dataset sourced from Huawei, reveal that HDLdebugger outperforms 13 cutting-edge LLM baselines, displaying exceptional effectiveness in HDL code debugging.
Boolean Satisfiability problems are vital components in Electronic Design Automation, particularly within the Logic Equivalence Checking process. Currently, SAT solvers are employed for these problems and neural network is tried as assistance to solvers. However, as SAT problems in the LEC context are distinctive due to their predominantly unsatisfiability nature and a substantial proportion of UNSAT-core variables, existing neural network assistance has proven unsuccessful in this specialized domain. To tackle this challenge, we propose IB-Net, an innovative framework utilizing graph neural networks and novel graph encoding techniques to model unsatisfiable problems and interact with state-of-the-art solvers. Extensive evaluations across solvers and datasets demonstrate IB-Net's acceleration, achieving an average runtime speedup of 5.0% on industrial data and 8.3% on SAT competition data empirically. This breakthrough advances efficient solving in LEC workflows.
In this paper, we continue the study on variance of the number of squarefull numbers in short intervals $(x, x + 2 \sqrt{x} H + H^2]$ with $X \le x \le 2X$. We obtain the expected asymptotic for this variance over the range $X^ε\le H \le X^{0.180688...}$ unconditionally and over the optimal range $X^ε\le H \le X^{0.25 - ε}$ conditionally on the Riemann Hypothesis or the Lindelöf Hypothesis.
Analogy-making between narratives is crucial for human reasoning. In this paper, we evaluate the ability to identify and generate analogies by constructing a first-of-its-kind large-scale story-level analogy corpus, \textsc{StoryAnalogy}, which contains 24K story pairs from diverse domains with human annotations on two similarities from the extended Structure-Mapping Theory. We design a set of tests on \textsc{StoryAnalogy}, presenting the first evaluation of story-level analogy identification and generation. Interestingly, we find that the analogy identification tasks are incredibly difficult not only for sentence embedding models but also for the recent large language models (LLMs) such as ChatGPT and LLaMa. ChatGPT, for example, only achieved around 30% accuracy in multiple-choice questions (compared to over 85% accuracy for humans). Furthermore, we observe that the data in \textsc{StoryAnalogy} can improve the quality of analogy generation in LLMs, where a fine-tuned FlanT5-xxl model achieves comparable performance to zero-shot ChatGPT.
Let $s_1, s_2, s_3, \cdots$ be the set of squarefree numbers in ascending order. In this paper, we prove that the following asymptotic on moments of gaps between squarefree numbers \[ \sum_{s_{k+1} \le x} (s_{k+1} - s_k)^γ\sim B(γ) x \; \; \mbox{ with some constant} \; \; B(γ) > 0 \] is true for $0 \le γ< 3.75$. This improves the previous best range $0 \le γ< 3.6875$.
In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0 < θ< 1/2$, almost all short intervals $(x, x + x^{1/2 + θ}]$ contain about $\frac{ζ(3/2)}{2 ζ(3)} x^θ$ squarefull numbers.
Abduction has long been seen as crucial for narrative comprehension and reasoning about everyday situations. The abductive natural language inference ($α$NLI) task has been proposed, and this narrative text-based task aims to infer the most plausible hypothesis from the candidates given two observations. However, the inter-sentential coherence and the model consistency have not been well exploited in the previous works on this task. In this work, we propose a prompt tuning model $α$-PACE, which takes self-consistency and inter-sentential coherence into consideration. Besides, we propose a general self-consistent framework that considers various narrative sequences (e.g., linear narrative and reverse chronology) for guiding the pre-trained language model in understanding the narrative context of input. We conduct extensive experiments and thorough ablation studies to illustrate the necessity and effectiveness of $α$-PACE. The performance of our method shows significant improvement against extensive competitive baselines.
Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerful numbers. We prove that its real logarithmic spectrum takes values from $- \sqrt{2} / (4 + \sqrt{2}) = -0.26160...$ to $1$ while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from $0$ to $1$. In the course of this study, we correct a proof of Granville and Soundararajan concerning contribution of small primes in the study of mean value of multiplicative functions.
We answer a number of questions of Erdős on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterlé.
In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_εN^{1/2 - ε}$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d \ll N^{1/2}$ unconditionally. We also prove some partial results when $k \ge 4$ and pose some open questions.