arXiv subjects

Hung Viet Chu

Publications and source records attributed to Hung Viet Chu.

At least 19 recordsLinked to original sources

arXiv2026

Generalizing a Pair of Diophantine Equations

For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\ =\ \frac{(a-1)(b-1)}{2} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer $m$ and its complement $ab-a-b-m$. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to $$ ax+by\ =\ \frac{(a-1)(b-1)}{k} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{k}, $$ where $k$ is a fixed positive integer. We then obtain explicit results when $a$ and $b$ are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when $b\equiv \pm1\mod a$, when $b$ is replaced by a higher power, and when the parameters are squared.

math.NT↗

arXiv2026

Counting Schreier Sets Under Neighborhood Conditions

We count Schreier sets that satisfy a neighborhood condition, including $k$-clustered, $k$-consecutive-free, $k$-neighbored, $k$-isolated, and closed under integral $2$-averages. For the first four conditions, we determine the initial counts and prove linear recurrence relations. For the last condition, we prove a recurrence that involves the divisor counting function.

math.CO↗

arXiv2026

Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers

A finite nonempty set $F\subset\mathbb{N}$ is Schreier if $\min F\ge |F|$. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If $\mathcal J_{k,n}$ is the collection of Schreier sets that are the union of exactly $k$ separated intervals, then the sequence $(|\mathcal{J}_{k,n}|)_{n=1}^\infty$ satisfies the characteristic polynomial $p_k(x) = (x-1)^{2k+1}(x+1)^k$. Furthermore, we introduce the new concept of $k$-super-Schreier sets and let $\mathcal{S}_{k,n}$ denote the collection of $k$-super Schreier sets whose maximum is $n$. We show that the sequence $(|\mathcal{S}_{k,n}|)_{n=1}^\infty$ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of $n$.

math.CO↗

arXiv2026

Schreier-Type Sets and Linear Recurrences: Connections and Developments

We demonstrate several common techniques for proving linear recurrences from counting Schreier-type sets. These techniques include formula-based arguments, bijective proofs, mathematical induction, the inclusion-exclusion principle, and the characteristic polynomial method. As new contributions, we examine symmetric maximal Schreier sets, Schreier sets that contain a prescribed integer, and Schreier sets that avoid integers belonging to a fixed arithmetic progression. Along the way, we employ useful techniques for identifying meaningful patterns in data and establishing technical identities. The results presented here, together with the diverse proof techniques employed, are expected to serve as a valuable resource for undergraduate researchers interested in this area.

math.CO↗

arXiv2026

Integers Having $F_{2k}$ in Both Zeckendorf And Chung-Graham Decompositions

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed Fibonacci numbers with coefficients $0,1$, or $2$ such that between two coefficients $2$, there is a coefficient $0$. We discover a correspondence between a lexicographically ordered sublist of Zeckendorf decompositions and letters in the golden string $\mathcal{S}$. Likewise, we identify a dual correspondence for Chung-Graham decompositions. We then use these correspondences to give the set of all positive integers having $F_{2k}$ in both of their Zeckendorf and Chung-Graham decompositions.

math.NT↗

arXiv2026

On a Pair of Diophantine Equations

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases.

math.NT↗

arXiv2026

Linear Recurrences of Generalized Schreier Sets Revisited

For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\ |\{F\subset\{1, 2, \ldots, n\}\,:\, q\min F\ge p|F|\mbox{ and }n\in F\}|.$$ Using the Inclusion-Exclusion Principle, Beanland et al. proved the recurrence $$|\mathcal{S}^{p/q}_{n}|\ =\ \sum_{k=1}^q(-1)^{k+1}\binom{q}{k}|\mathcal{S}^{p/q}_{n-k}| + |\mathcal{S}^{p/q}_{n-(p+q)}|.$$ We show that $(|\mathcal{S}^{p/q}_n|)_{n=1}^\infty$ is a subsequence with terms taken periodically from Padovan-like sequences which satisfy simple recurrence relations. As an application, we obtain an alternative proof of the above linear recurrence. Furthermore, a similar result holds for the sequence $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$ that counts maximal $(p,q)$-Schreier sets. We end with a discussion of the relation between $(|\mathcal{S}^{p/q}_{n}|)_{n=1}^\infty$ and $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$.

math.CO↗

arXiv2025

Visualize Geometric Series

We review Mabry's, Edgar's, and the Viewpoints 2000 Group's proofs without words for the geometric series formula. Mabry and Edgar proved without words that $$\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \cdots\ =\ \frac{3}{4}\quad\mbox{ and }\quad\frac{4}{9} + \left(\frac{4}{9}\right)^2 + \left(\frac{4}{9}\right)^3 + \cdots\ =\ \frac{4}{5},$$ respectively. We show that their proofs satisfy certain requirements that make them unique. We then illustrate a common idea between their and the Viewpoints 2000 Group's proofs.

math.HO↗

arXiv2025

Lower-Order Refinements of Greedy Approximation

For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and 3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic. If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$).

math.FA↗

arXiv2025

Problems Regarding a Pair of Diophantine Equations

For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative integral solution $(x, y)$. Furthermore, the solution is unique. In this note, we summarize recent results and some new ones on the solution of the two equations and provide an overview of problems for future investigation, some of which were presented at the 2025 International Conference on Class Groups of Number Fields and Related Topics.

math.NT↗

arXiv2025

Enlarge Greedy Sums in Greedy-Type Properties by Different Factors

It was previously known that the almost greedy (AG) property essentially remains the same when we enlarge greedy sums in the classical definition by a factor $λ\geqslant 1$. The present paper shows that if instead, we enlarge greedy sums in a reformulation of the AG property, we obtain a weaker one. However, the new property is essentially independent of the enlarging factor $λ$ once $λ> 1$. In contrast, we observe a continuum of partially greedy-like properties by varying $λ\in [1,\infty)$. Last but not least, under a threshold for $λ$, we characterize the isometric version of the weakened AG property. Specifically, the characterization holds if and only if $λ\in [1, 2]$.

math.FA↗

arXiv2025

A Pair of Diophantine Equations and Fibonacci-Like Sequences

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.

math.NT↗

arXiv2025

Linear Recurrences from Counting Schreier-Type Multisets

A nonempty set $F$ is Schreier if $\min F\ge |F|$. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have been discovered. Building on these works, we prove a linear recurrence for the sequence that counts multisets $F$ with $\min F\ge p|F|$. In particular, if we let $$\mathcal{A}^{(s)}_{p, n}\ :=\ \{F\subset \{\underbrace{1, \ldots, 1}_{s}, \ldots, \underbrace{n-1, \ldots, n-1}_{s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ then $$|\mathcal{A}^{(s)}_{p, n}| = \sum_{i=0}^s|\mathcal{A}^{(s)}_{p, n-1-ip}|.$$ If we color $s$ copies of the same integer by different colors from $1$ to $s$, i.e., $\mathcal{B}^{(s)}_{p, n}:= $ $$\{F\subset \{1_{1}, \ldots, 1_{s}, \ldots, (n-1)_1, \ldots, (n-1)_{s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ then $$|\mathcal{B}^{(s)}_{p, n}| = \sum_{i=0}^s \binom{s}{i}| \mathcal{B}^{(s)}_{p, n-1-ip}|.$$ Lastly, we count Schreier sets that do not admit multiples of a given integer $u\ge 2$ and witness linear recurrences whose coefficients are drawn from the $u$th row of the Pascal triangle and have alternating signs, except possibly the last one.

math.CO↗

arXiv2025

Lebesgue-type estimates for greedy algorithms in quasi-Banach spaces

We continue the study of Lebesgue-type parameters for various greedy algorithms in quasi-Banach spaces. First, we introduce a parameter that can be used with the quasi-greedy parameter to obtain the exact growth of the Lebesgue parameter for strong partially greedy bases. Second, we establish a new upper bound for the Lebesgue parameter for semi-greedy bases using the quasi-greedy and the squeeze symmetry parameters. Finally, we answer several open questions regarding the optimal power in various bounds proved in [F. Albiac, J. L. Ansorena, and P. M. Berná, New parameters and Lebesgue-type estimates in greedy approximation, Forum Math. Sigma 10 (2022), 1-39].

math.FA↗

arXiv2025

On sequential greedy-type bases

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of $\mathbb{N}$. In this paper, we fix a sequence $(a_n)_{n=1}^\infty$ and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the $\mathcal{F}_{(a_n)}$-almost greedy property. Our first result shows that the $\mathcal{F}_{(a_n)}$-almost greedy property is equivalent to the classical almost greedy property if and only if $(a_n)_{n=1}^\infty$ is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence $(a_n)_{n=1}^\infty$, we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties.

math.FA↗

arXiv2025

Squaring the Circle Revisited

Squaring the circle is impossible, but it can be squared approximately. Ramanujan gave a construction correct to eight decimal places. In his book Mathographics, Dixon gave constructions correct to three decimal places. In this article, we provide a new construction correct to three decimal places and another correct to nine decimal places.

math.GM↗

arXiv2024

Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients $2$, there must be a coefficient $0$. This paper characterizes all positive integers that do not have $F_{2k}$ ($k\ge 1$) in their decompositions. This continues the work of Kimberling, Carlitz et al., Dekking, and Griffiths, to name a few, who studied such a characterization for Zeckendorf decomposition.

math.GM↗

arXiv2024

Composite Numbers in an Arithmetic Progression

One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those able to go faster, an instructor must be prepared to give problems of different difficulty levels. Using Dirichlet's Theorem as a case study, we create and discuss a family of problems in number theory that highlight the relative strengths and weaknesses of different ways to approach a question and show how to invite students to extend the problems and explore research-level mathematics.

math.HO↗