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Tsz Ho Chan

Publications and source records attributed to Tsz Ho Chan.

At least 37 records · Page 2Linked to original sources
arXiv2022

A note on powerful numbers in short intervals

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{\log^{1+ε} y})$ for squarefull numbers and the bound $O(y^{(2 + ε)/k})$ for $k$-full numbers when $k \ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numbers.

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arXiv2020

On the critical exponent for $k$-primitive sets

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-λ}$ over a primitive set is maximized by the primes if and only if $λ$ is at least the critical exponent $τ_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. One may similarly consider the critical exponent $τ_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that $τ_2 < 0.8$, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that $τ_k$ tends to zero as $k\to\infty$.

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arXiv2020

A generalization of primitive sets and a conjecture of Erdős

A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.

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arXiv2019

Gaps between divisible terms in $a^2 (a^2 + 1)$

Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \gg a (\log a)^{1/8} / (\log \log a)^{12}$. We also obtain $b \gg_εa^{15/14 - ε}$ under the abc conjecture.

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arXiv2015

Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue

In this paper, we study how small a box contains at least two points from a modular hyperbola $x y \equiv c \pmod p$. There are two such points in a square of side length $p^{1/4 + ε}$. Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6 + ε}$ or the least quadratic nonresidue is less than $p^{1/(6 \sqrt{e}) + ε}$.

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arXiv2015

Shortest Distance in Modular Cubic Polynomials

In this paper, we study how small a box contains at least two points from a modular cubic polynomial $y \equiv a x^3 + b x^2 + c x + d \pmod p$ with $(a, p) = 1$. We prove that some square of side length $p^{1/6 + ε}$ contains two such points.

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arXiv2015

Finding almost squares VI

In this paper, we continue the study of almost squares and extend the result of the author's fourth paper of the series to almost squares with closer factors.

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arXiv2014

On sums of Ramanujan sums

Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$

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arXiv2014

Factors of almost squares and lattice points on circles

In this paper, we consider a conjecture of Erdos and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.

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arXiv2013

Numbers with three close factorizations

In this paper, we study numbers $n$ that can be factored in three different ways as $n = A_1 B_1 = A_2 B_2 = A_3 B_3$ where $A_1$, $A_2$, $A_3$ are close to each other and $B_1$, $B_2$, $B_3$ are close to each other.

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arXiv2009

Additive bases arising from functions in a Hardy field

A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.

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arXiv2008

An almost all result on $q_1 q_2 \equiv c \pmod q$

In this paper we consider the congruence equation $q_1 q_2 \equiv c \pmod q$ with $a < q_1 \leq a + q^{1/2+ε}$ and $b < q_2 \leq b + q^{1/2+ε}$ and show that it has solution for almost all $a$ and $b$. Then we apply it to a question of Fujii and Kitaoka as well as generalize it to more variables. At the end, we will present a new way to attack the above congruence equation question through higher moments.

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arXiv2007

Finding Almost Squares III

An almost square of type 2 is an integer $n$ that can be factored in two different ways as $n = a_1 b_1 = a_2 b_2$ with $a_1$, $a_2$, $b_1$, $b_2 \approx \sqrt{n}$. In this paper, we shall improve upon previous result on short intervals containing an almost square of type 2. This leads to an inquiry of finding a short interval around $x$ that contains an integer divisible by some integer in $[x^c, 2 x^c]$ with $0 < c < 1$.

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arXiv2007

Approximating reals by sums of rationals

We consider the question of approximating any real number $α$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator.

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