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Hung Viet Chu

Publications and source records attributed to Hung Viet Chu.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2024

A Pair of Diophantine Equations Involving the Fibonacci Numbers

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique. Let $F_n$ be the $n$th Fibonacci number. When $(a,b) = (F_n, F_{n+1})$, it is known that there is an explicit formula for the unique solution $(x,y)$. We establish formulas to compute the solution when $(a,b) = (F_n^2, F_{n+1}^2)$ and $(F_n^3, F_{n+1}^3)$, giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution.

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arXiv2024

Weighted Schreier-type Sets and the Fibonacci Sequence

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $ω_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } ω_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ First, we prove that $$a_{k,k+\ell} \ =\ 2F_{k+\ell},\mbox{ for all }\ell\geqslant 0\mbox{ and }k\geqslant \ell+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > ω_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where $ω_{2,3}(E) = \sum_{i\in E, i\neq 2, 3}1$.

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arXiv2024

Strong Partially Greedy Bases with respect to An Arbitrary Sequence

For Schauder bases, Dilworth et al. introduced and characterized the partially greedy property, which is strictly weaker than the (almost) greedy property. Later, Berasategui et al. defined and studied the strong partially greedy property for general bases. Let $\mathbf n$ be any strictly increasing sequence of positive integers. In this paper, we define the strong partially greedy property with respect to $\mathbf n$, called the ($\mathbf n$, strong partially greedy) property. We give characterizations of this new property, study relations among ($\mathbf n$, strong partially greedy) properties for different sequences $\mathbf n$, establish Lebesgue-type inequalities for the ($\mathbf n$, strong partially greedy) parameter, investigate ($\mathbf n$, strong partially greedy) bases with gaps, and weighted ($\mathbf n$, strong partially greedy) bases, to name a few. Furthermore, we introduce the ($\mathbf n$, almost greedy) property and equate the property to a strengthening of the ($\mathbf n$, strong partially greedy) property. This paper can be viewed both as a survey of recent results regarding strong partially greedy bases and as an extension of these results to an arbitrary sequence instead of $\mathbb{N}$.

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arXiv2024

Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

We prove that for every countable ordinal $ξ$, the Tsirelson's space $T_ξ$ of order $ξ$, is naturally, i.e., via the identity, $3$-isomorphc to its modified version. For the first step, we prove that the Schreier family $\mathcal{S}_ξ$ is the same as its modified version $ \mathcal{S}^M_ξ$, thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on $T_ξ$ has $2^{\mathfrak c}$ closed ideals.

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arXiv2024

A Threshold for the Best Two-term Underapproximation by Egyptian Fractions

Let $\mathcal{G}$ be the greedy algorithm that, for each $θ\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = θ$. For natural numbers $p < q$, let $Υ(p,q)$ denote the smallest positive integer $j$ such that $p$ divides $q+j$. Continuing Nathanson's study of two-term underapproximations, we show that whenever $Υ(p,q) \leqslant 3$, $\mathcal{G}$ gives the (unique) best two-term underapproximation of $p/q$; i.e., if $1/x_1 + 1/x_2 < p/q$ for some $x_1, x_2\in \mathbb{N}$, then $1/x_1 + 1/x_2 \leqslant 1/a_1+1/a_2$. However, the same conclusion fails for every $Υ(p,q)\geqslant 4$. Next, we study stepwise underapproximation by $\mathcal{G}$. Let $e_{m} = θ- \sum_{n=1}^{m}1/a_n$ be the $m$th error term. We compare $1/a_m$ to a superior underapproximation of $e_{m-1}$, denoted by $N/b_m$ ($N \in\mathbb{N}_{\geqslant 2}$), and characterize when $1/a_m = N/b_m$. One characterization is $a_{m+1} \geqslant N a_m^2 - a_m + 1$. Hence, for rational $θ$, we only have $1/a_m = N/b_m$ for finitely many $m$. However, there are irrational numbers such that $1/a_m = N/b_m$ for all $m$. Along the way, various auxiliary results are encountered.

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arXiv2023

On Schreier-type Sets, Partitions, and Compositions

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in increasing order are non-decreasing. This note establishes a connection between sparse Schreier-type sets and (restricted) partition numbers. One of our results states that if $\mathcal{G}_{n,\ell}$ consists of partitions of $n$ that contain no parts in $\{2, \ldots, \ell\}$, and \begin{equation*} \mathcal{A}_{n,\ell} \ :=\ \{A\subset \{1, \ldots, n\}\,:\, n\in A, A\mbox{ is sparse and }\ell\mbox{-strong Schreier}\}, \end{equation*} then $$|\mathcal{A}_{n,\ell}|\ =\ |\mathcal{G}_{n-1,\ell}|, \quad n, \ell\in \mathbb{N}.$$ The special case $\mathcal{G}_{n-1, 1}$ consists of all partitions of $n-1$. Besides partitions, integer compositions are also investigated.

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arXiv2023

Larger Greedy Sums for Reverse Partially Greedy Bases

An interesting result due to Dilworth et al. was that if we enlarge greedy sums by a constant factor $λ> 1$ in the condition defining the greedy property, then we obtain an equivalence of the almost greedy property, a strictly weaker property. Previously, the author of the present paper showed that enlarging greedy sums by $λ$ in the condition defining the partially greedy (PG) property also strictly weakens the property. However, enlarging greedy sums in the definition of reverse partially greedy (RPG) bases by Dilworth and Khurana again gives RPG bases. The companion of PG and RPG bases suggests the existence of a characterization of RPG bases which, when greedy sums are enlarged, gives an analog of a result that holds for partially greedy bases. In this paper, we show that such a characterization indeed exists, answering positively a question previously posed by the author.

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arXiv2023

Schreier Multisets and the $s$-step Fibonacci Sequences

Inspired by the surprising relationship (due to A. Bird) between Schreier sets and the Fibonacci sequence, we introduce Schreier multisets and connect these multisets with the $s$-step Fibonacci sequences, defined, for each $s\geqslant 2$, as: $F^{(s)}_{2-s} = \cdots = F^{(s)}_0 = 0$, $F^{(s)}_1 = 1$, and $F^{(s)}_{n} = F^{(s)}_{n-1} + \cdots + F^{(s)}_{n-s}, \mbox{ for } n\geqslant 2$. Next, we use Schreier-type conditions on multisets to retrieve a family of sequences which satisfy a recurrence of the form $a(n) = a(n-1) + a(n-u)$, with $a(n) = 1$ for $n = 1,\ldots, u$. Finally, we study nonlinear Schreier conditions and show that these conditions are related to integer decompositions, each part of which is greater than the number of parts raised to some power.

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arXiv2023

Approximation by Egyptian Fractions and the Weak Greedy Algorithm

Let $0 < θ\leqslant 1$. A sequence of positive integers $(b_n)_{n=1}^\infty$ is called a weak greedy approximation of $θ$ if $\sum_{n=1}^{\infty}1/b_n = θ$. We introduce the weak greedy approximation algorithm (WGAA), which, for each $θ$, produces two sequences of positive integers $(a_n)$ and $(b_n)$ such that a) $\sum_{n=1}^\infty 1/b_n = θ$; b) $1/a_{n+1} < θ- \sum_{i=1}^{n}1/b_i < 1/(a_{n+1}-1)$ for all $n\geqslant 1$; c) there exists $t\geqslant 1$ such that $b_n/a_n \leqslant t$ infinitely often. We then investigate when a given weak greedy approximation $(b_n)$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $(a_n)$ with $a_1\geqslant 2$ and $a_n\rightarrow\infty$, there exist $θ$ and $(b_n)$ such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence $(a_n)$. Finally, we address the uniqueness of $θ$ and $(b_n)$ and apply our framework to specific sequences.

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arXiv2023

The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded by Certain Functions $f$

Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a function $f$ in a large collection $\mathcal{F}$ (closed under composition), we define and characterize $f$-greedy and $f$-almost greedy bases. We study relations among these bases as $f$ varies and show that while a basis is not almost greedy, it can be $f$-greedy for some $f\in \mathcal{F}$. Furthermore, we prove that for all non-identity function $f\in \mathcal{F}$, we have the surprising equivalence $$f\mbox{-greedy}\ \Longleftrightarrow \ f\mbox{-almost greedy}.$$ We give various examples of Banach spaces to illustrate our results.

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arXiv2023

Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to $1$. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be $1$-suppression unconditional, thus settling Problem 4.4 from [F. Albiac and J.L. Ansorena, Characterization of $1$-almost greedy bases, Rev. Mat. Complut. 30 (2017), no. 1, 13-24]. In particular, our construction demonstrates that bases with Property (A) need not be $1$-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

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arXiv2023

On Approximation Spaces and Greedy-type Bases

The purpose of this paper is to introduce $ω$-Chebyshev-greedy and $ω$-partially greedy approximation classes and to study their relation with $ω$-approximation spaces, where the latter are a generalization of the classical approximation spaces. The relation gives us sufficient conditions of when certain continuous embeddings imply different greedy-type properties. Along the way, we generalize a result by P. Wojtaszczyk as well as characterize semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous result by the first author.

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arXiv2023

On consecutive greedy and other greedy-like type of bases

We continue our study of the Thresholding Greedy Algorithm when we restrict the vectors involved in our approximations so that they either are supported on intervals of $\mathbb N$ or have constant coefficients. We introduce and characterize what we call consecutive greedy bases and provide new characterizations of almost greedy and squeeze symmetric Schauder bases. Moreover, we investigate some cases involving greedy-like properties with constant 1 and study the related notion of Property (A,$τ$).

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arXiv2023

Performance of the Thresholding Greedy Algorithm with Larger Greedy Sums

The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $λ\geqslant 1$. We introduce the so-called $λ$-almost greedy and $λ$-partially greedy bases. The case when $λ= 1$ gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is $λ$-almost greedy for all (some) $λ\geqslant 1$. However, for each $λ> 1$, there exists an unconditional basis that is $λ$-partially greedy but is not $1$-partially greedy. Furthermore, we investigate and give examples when a basis is 1. not almost greedy with constant $1$ but is $λ$-almost greedy with constant $1$ for some $λ> 1$, and 2. not strong partially greedy with constant $1$ but is $λ$-partially greedy with constant $1$ for some $λ> 1$. Finally, we prove various characterizations of different greedy-type bases.

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arXiv2023

On Weighted Greedy-type Bases

In this paper, we study weights for the Thresholding Greedy Algorithm (TGA). While previous work focused on sequential weights $ς= (s_n)_{n\in\mathbb{N}}$ on each positive integer, we study a more general weight $ω= (w_A)_{A\subset\mathbb{N}}$ on each set $A\subset \mathbb{N}$. We define and characterize $ω$-(almost) greedy bases. Furthermore, we leverage existing results to show that there exists an $ω$-greedy unconditional basis that is not $ς$-almost greedy for any weight sequence $ς$. Last but not least, we show the equivalence between $ω$-semi-greedy bases and $ω$-almost greedy bases when $ω$ is a so-called structured weight, thus considerably extending the equivalence previously known to hold for sequential weights.

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arXiv2023

Extensions and new characterizations of some greedy-type bases

Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.

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arXiv2022

Schreier families and $\mathcal{F}$-(almost) greedy bases

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is called $\mathcal{F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb{N}$, and $G_m(x)$, we have $$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ is the scalar field. From the definition, any $\mathcal{F}$-greedy basis is quasi-greedy and so, the notion of being $\mathcal{F}$-greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal{F}$-greedy bases as being $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal{F}$-almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal{F}$-greedy. For a countable ordinal $α$, we consider the case $\mathcal{F}=\mathcal{S}_α$, where $\mathcal{S}_α$ is the Schreier family of order $α$. We show that for each $α$, there is a basis that is $\mathcal{S}_α$-greedy but is not $\mathcal{S}_{α+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $α< β$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_α\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_β\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$

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