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Hung Viet Chu

Publications and source records attributed to Hung Viet Chu.

At least 55 records · Page 3Linked to original sources
arXiv2022

Linear recurrences of order at most two in nontrivial small divisors and large divisors

For each positive integer $N$, define $$S'_N \ =\ \{1 < d < \sqrt{N}: d|N\}\mbox{ and }L'_N \ =\ \{\sqrt{N} < d < N : d|N\}.$$ Recently, Chentouf characterized all positive integers $N$ such that the set of small divisors $\{d\le \sqrt{N}: d|N\}$ satisfies a linear recurrence of order at most two. We nontrivially extend the result by excluding the trivial divisor $1$ from consideration, which dramatically increases the analysis complexity. Our first result characterizes all positive integers $N$ such that $S'_N$ satisfies a linear recurrence of order at most two. Moreover, our second result characterizes all positive $N$ such that $L'_N$ satisfies a linear recurrence of order at most two, thus extending considerably a recent result that characterizes $N$ with $L'_N$ being in an arithmetic progression.

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arXiv2022

A Note on the Fibonacci Sequence and Schreier-type Sets

A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$), where $$\mathcal{K}_{n, p, q} \ = \ \{A\subset \{1, \ldots, n\}\,:\, \mbox{either }A = \emptyset \mbox{ or } (\max A-\max_2 A = p\mbox{ and }\min A\ge |A|\ge q)\}$$ and $\max_2 A$ is the second largest integer in $A$, given that $|A|\ge 2$. When $p = 1$ and $q=2$, we have that $(|\mathcal{K}_{n, 1, 2}|)_{n=1}^\infty$ is the Fibonacci sequence. As a corollary, we obtain a new combinatorial interpretation for the sequence $(F_n + n)_{n=1}^\infty$.

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arXiv2022

Additive complements for two given asymptotic densities

Let $0\le α\le β\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = α$ and $\bar{d}(A+B) = β$, where $\underline{d}(A+ B)$ and $\bar{d}(A+B)$ are the lower and upper asymptotic densities of the set $A+B$, respectively. This partially answers a question by Faisant et al. A theorem involving the so-called highly sparse sets was proved in the previous arXiv version of this note; however, as pointed out by Sai Teja Somu, the proof of the theorem was flawed. The theorem is now an open question.

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arXiv2022

On some characterizations of greedy-type bases

In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding Greedy Algorithm. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak Thresholding Greedy Algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.

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arXiv2022

On a Relation between Schreier-type Sets and a Modification of Turán Graphs

Recently, a relation between Schreier-type sets and Turán graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for $p, q\ge 1$, let $$\mathcal{A}_q := \{F\subset\mathbb{N}: |F| = 1 \mbox{ or }F\mbox{ is an arithmetic progression with difference } q\}$$ and $$Sr(n, p, q)\ :=\ \#\{F\subset \{1, \ldots, n\}\,:\, p\min F\ge |F|\mbox{ and }F\in \mathcal{A}_q\}.$$ We show that $$Sr(n, p, q) \ =\ T(n+1, pq+1, q),$$ where $T(\cdot, \cdot, \cdot)$ is the number of edges of an $n$-vertex graph that is a modification of Turán graphs. We also prove that $Sr(n,p,q)$ is the partial sum of certain sequences.

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arXiv2022

Variations of Property (A) Constants and Lebesgue-type Inequalities for the Weak Thresholding Greedy Algorithms

Albiac and Wojtaszczyk introduced property (A) to characterize $1$-greedy bases. Later, Dilworth et al. generalized the concept to $C$-property (A), where the case $C = 1$ gives property (A). They (among other results) characterized greedy bases by unconditionality and $C$-property (A). In this paper, we extend the definition of the so-called A-property constant to (A,$τ$)-property constants and use the extension to obtain new estimates for various Lebesgue parameters. Furthermore, we study the relation among (A,$τ$)-property constants and other well-known constants when $τ$ varies.

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arXiv2021

Generalized Schreier sets, linear recurrence relation, Turán graphs

We prove a linear recurrence relation for a large family of generalized Schreier sets, which generalizes the Fibonacci recurrence proved by Bird and higher order Fibonacci recurrence proved by the second author et al. Furthermore, we show a relationship between Schreier-type sets and Turán graphs.

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arXiv2021

On Arithmetic Progressions of Powers in Cyclotomic Polynomials

We determine necessary conditions for when powers corresponding to positive/negative coefficients of $Φ_{n}$ are in arithmetic progression. When $n = pq$ for any primes $q>p>2$, our conditions are also sufficient. Finally, we generalize the result when $n = pq$ to the so-called inclusion-exclusion polynomials first introduced by Bachman.

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arXiv2021

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding the trivial divisors $1$ and $\sqrt{n}$ (when $n$ is a square). Surprisingly, the length of our arithmetic progression cannot exceed $5$.

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arXiv2021

Partial Sums of the Fibonacci Sequence

Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is the resulting sequence from applying $P$ to $(F_n)$ $k$ times. Second, we provide a combinatorial interpretation of the numbers in $P^k(F_n)$.

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arXiv2020

The Fibonacci Sequence and Schreier-Zeckendorf Sets

A finite subset of the natural numbers is weak-Schreier if $\min S \ge |S|$, strong-Schreier if $\min S>|S|$, and maximal if $\min S = |S|$. Let $M_n$ be the number of weak-Schreier sets with $n$ being the largest element and $(F_n)_{n\geq -1}$ denote the Fibonacci sequence. A finite set is said to be Zeckendorf if it does not contain two consecutive natural numbers. Let $E_n$ be the number of Zeckendorf subsets of $\{1,2,\ldots,n\}$. It is well-known that $E_n = F_{n+2}$. In this paper, we first show four other ways to generate the Fibonacci sequence from counting Schreier sets. For example, let $C_n$ be the number of weak-Schreier subsets of $\{1,2,\ldots,n\}$. Then $C_n = F_{n+2}$. To understand why $C_n = E_n$, we provide a bijective mapping to prove the equality directly. Next, we prove linear recurrence relations among the number of Schreier-Zeckendorf sets. Lastly, we discover the Fibonacci sequence by counting the number of subsets of $\{1,2,\ldots, n\}$ such that two consecutive elements in increasing order always differ by an odd number.

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arXiv2020

Higher Order Fibonacci Sequences from Generalized Schreier sets

A Schreier set $S$ is a subset of the natural numbers with $\min S\ge |S|$. It has been known that the sequence $(a_{1,n})$, where $$a_{1,n}\ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|\}|,$$ is the Fibonacci sequence. Generalizing this result, we prove that for all $p\in \mathbb{N}$, the sequence $(a_{p,n})$, where $$a_{p, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S\ge p|S|\}|,$$ has a linear recurrence relation of higher order. We investigate further by requiring that ${\rm min}_2 S\ge q |S|$, where $\min_2 S$ is the second smallest element of $S$. We prove a linear recurrence relation for the sequence $(a_{p, q, n})$, where $$a_{p, q, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n, \min S \ge p|S|\mbox{ and } {\rm min}_2 S\ge q|S|\}|,$$ and discuss a curious relationship between $(a_{q, n})$ and $(a_{p, q, n})$.

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arXiv2020

Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let $L(n)$ be the number of summands in the partition of $n$. We characterize all positive integers such that $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) = L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n) < L(n+1) > L(n+2)$ and a divot of $L$ if $L(n) > L(n+1) < L(n+2)$. We characterize all such peaks and divots of $L$.

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arXiv2020

On Generalized Zeckendorf Decompositions and Generalized Golden Strings

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. A natural generalization of this theorem is to look at the sequence defined as follows: for $n\ge 2$, let $F_{n,1} = F_{n,2} = \cdots = F_{n,n} = 1$ and $F_{n, m+1} = F_{n, m} + F_{n, m+1-n}$ for all $m\ge n$. It is known that every positive integer has a unique representation as a sum of $F_{n,m}$'s where the indexes of summands are at least $n$ apart. We call this the $n$-decomposition. Griffiths showed an interesting relationship between the Zeckendorf decomposition and the golden string. In this paper, we continue the work to show a relationship between the $n$-decomposition and the generalized golden string.

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arXiv2020

On Even Perfect Numbers II

Let $k>2$ be a prime such that $2^k-1$ is a Mersenne prime. Let $n = 2^{α-1}p$, where $α>1$ and $p<3\cdot 2^{α-1}-1$ is an odd prime. Continuing the work of Cai et al. and Jiang, we prove that $n\ |\ σ_k(n)$ if and only if $n$ is an even perfect number $\neq 2^{k-1}(2^k-1)$. Furthermore, if $n = 2^{α-1}p^{β-1}$ for some $β>1$, then $n\ |\ σ_5(n)$ if and only if $n$ is an even perfect number $\neq 496$.

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arXiv2020

Sets of Cardinality 6 Are Not Sum-dominant

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. Hegarty used a nontrivial algorithm to find that $8$ is the smallest cardinality of a sum-dominant set. Since then, Nathanson has asked for a human-understandable proof of the result. However, due to the complexity of the interactions among numbers, it is still questionable whether such a proof can be written down in full without computers' help. In this paper, we present a computer-free proof that a sum-dominant set must have at least $7$ elements. We also answer the question raised by the author of the current paper et al about the smallest sum-dominant set of primes, in terms of its largest element. Using computers, we find that the smallest sum-dominant set of primes has $73$ as its maximum, smaller than the value found before.

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arXiv2020

Union of Two Arithmetic Progressions with the Same Common Difference Is Not Sum-dominant

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. We prove the following results. 1) The union of two arithmetic progressions (with the same common difference) is not sum-dominant. This result partially proves a conjecture proposed by the author in a previous paper; that is, the union of any two arbitrary arithmetic progressions is not sum-dominant. 2) Hegarty proved that a sum-dominant set must have at least $8$ elements with computers' help. The author of the current paper provided a human-verifiable proof that a sum-dominant set must have at least $7$ elements. A natural question is about the largest cardinality of sum-dominant subsets of an arithmetic progression. Fix $n\ge 16$. Let $N$ be the cardinality of the largest sum-dominant subset(s) of $\{0,1,\ldots,n-1\}$ that contain(s) $0$ and $n-1$. Then $n-7\le N\le n-4$; that is, from an arithmetic progression of length $n\ge 16$, we need to discard at least $4$ and at most $7$ elements (in a clever way) to have the largest sum-dominant set(s). 3) Let $R\in \mathbb{N}$ have the property that for all $r\ge R$, $\{1,2,\ldots,r\}$ can be partitioned into $3$ sum-dominant subsets, while $\{1,2,\ldots,R-1\}$ cannot. Then $24\le R\le 145$. This result answers a question by the author et al. in another paper on whether we can find a stricter upper bound for $R$.

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