arXiv subjects

Hung Viet Chu

Publications and source records attributed to Hung Viet Chu.

65 records · Page 4Linked to original sources

arXiv2020

On Sets with More Products than Quotients

Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\ &A - A \ =\ \{a_i - a_j\,|\,a_i,a_j\in A\}.\end{align*} The set $A$ is said to be MPTQ (more product than quotient) if $|A\cdot A|>|A/A|$ and MSTD (more sum than difference) if $|A + A|>|A - A|$. Since multiplication and addition are commutative while division and subtraction are not, it is natural to think that MPTQ and MSTD sets are very rare. However, they do exist. This paper first shows an efficient search for MPTQ subsets of $\{1,2,\ldots,n\}$ and proves that as $n\rightarrow \infty$, the proportion of MPTQ subsets approaches $0$. Next, we prove that MPTQ sets of positive numbers must have at least $8$ elements, while MPTQ sets of both negative and positive numbers must have at least $5$ elements. Finally, we investigate several sequences that do not have MPTQ subsets.

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arXiv2020

Representation of $\frac{1}{2}(F_n-1)(F_{n+1}-1)$ and $\frac{1}{2}(F_n-1)(F_{n+2}-1)$

Let $a, b\in \mathbb{N}$ be relatively prime. We consider $(a-1)(b-1)/2$, which arises in the study of the $pq$-th cyclotomic polynomial, where $p,q$ are distinct primes. We prove two possible representations of $(a-1)(b-1)/2$ as nonnegative, integral linear combinations of $a$ and $b$. Surprisingly, for each pair $(a,b)$, only one of the two representations exists and the representation is also unique. We then investigate the representations of $(F_n-1)(F_{n+1}-1)/2$ and $(F_n-1)(F_{n+2}-1)/2$, where $F_i$ is the $i^{th}$ Fibonacci number, and observe several nice patterns.

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arXiv2020

A Twist of a Ramanujan Identity

Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 + \frac{1}{11}\right) \left(1 + \frac{1}{19}\right), \end{align*} on which Berndt asked "Is this an isolated result, or are there other identities of this type?". Rebák provided formulas that generate many similar identities and believed that the curious identity is related to the reciprocal of the Landau-Ramanujan constant. In a previous work, Chu examined necessary and sufficient conditions for the integers in the identity and proved that there are only finitely many such identities. In this note, we twist the identity to have infinitely many Ramanujan-type identities.

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arXiv2020

A New Transcendental Number from $N^N$

We first give a summary of the history of transcendental numbers then use a nice technique by G. Dresden to prove a new transcendental number. In particular, while previous work looked at the last non-zero digit of $n^n$, we consider the digit right before its last non-zero digit and show that the infinite decimal built from these digits is transcendental.

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arXiv2020

On a Curious Identity of Ramanujan

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arXiv2019

When the Large Divisors of a Natural Number Are in Arithmetic Progression

Iannucci considered the positive divisors of a natural number $n$ that do not exceed the square root of $n$ and found all numbers whose such divisors are in arithmetic progression. Continuing the work, we define large divisors to be divisors at least $\sqrt{n}$ and find all numbers whose large divisors are in arithmetic progression. The asymptotic formula for the count of these numbers up to a bound $x$ is observed to be $\frac{x\log\log x}{\log x}$.

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arXiv2019

Infinite Families of Partitions into MSTD Subsets

A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{1,2,\ldots ,r\}$ as $r\rightarrow\infty$. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of $\{1,2,\ldots,r\}$ into two MSTD subsets as $r\rightarrow\infty$. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition $\{1,2,\ldots,r\}$ (for $r$ sufficiently large) into $k \ge 2$ MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such $k$. Next, let $R(k)$ be the smallest integer such that for all $r\ge R(k)$, $\{1,2,\ldots,r\}$ can be $k$-decomposed into MSTD subsets. We establish rough lower and upper bounds for $R(k)$. Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of $\{1,2,\ldots,r\}$ into $k$ MSTD subsets as $r\rightarrow \infty$.

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arXiv2019

When Sets Are Not Sum-dominant

Given a set $A$ of nonnegative integers, 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 $|A+A|>|A-A|$. In answering a question by Nathanson, Hegarty used a clever algorithm to find that the smallest cardinality of a sum-dominant set is $8$. Since then, Nathanson has been asking for a human-understandable proof of the result. We offer a computer-free proof that a set of cardinality less than $6$ is not sum-dominant. Furthermore, we prove that the introduction of at most two numbers into a set of numbers in an arithmetic progression does not give a sum-dominant set. This theorem eases several of our proofs and may shed light on future work exploring why a set of cardinality $6$ is not sum-dominant. Finally, we prove that if a set contains a certain number of integers from a specific sequence, then adding a few arbitrary numbers into the set does not give a sum-dominant set.

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arXiv2019

Generalizations of a Curious Family of MSTD Sets Hidden By Interior Blocks

A set $A$ is MSTD (more-sum-than-difference) or sum-dominant if $|A+A|>|A-A|$, and is RSD (restricted-sum dominant) if $|A\hat{+}A|>|A-A|$, where $A\hat{+}A$ is the set of sums of distinct elements in $A$. We study an interesting family of MSTD sets that have appeared many times in the literature (see the works of Hegarty, Martin and O'Bryant, and Penman and Wells). While these sets seem at first glance to be ad hoc, looking at them in the right way reveals a nice common structure. In particular, instead of viewing them as explicitly written sets, we write them in terms of differences between two consecutive numbers in increasing order. We denote this family by $\mathcal{F}$ and investigate many of its properties. Using $\mathcal{F}$, we are able to generate many sets $A$ with high value of $\log|A+A|/\log|A-A|$, construct sets $A$ with a fixed $|A+A|-|A-A|$ more economically than previous authors, and improve the lower bound on the proportion of RSD subsets of $\{0,1,2,\dots,n-1\}$ to about $10^{-25}$ (the previous best bound was $10^{-37}$). Lastly, by exhaustive computer search, we find six RSD sets with cardinality $15$, which is one lower than the smallest cardinality found to date, and find that $30$ is the smallest diameter of RSD sets.

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arXiv2019

The Schreier Space Does Not Have the Uniform $λ$-property

The $λ$-property and the uniform $λ$-property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the $λ$-property and asked if it has the uniform $λ$-property. In this paper, we show that Schreier space does not have the uniform $λ$-property. Furthermore, we show that the dual of the Schreier space does not have the uniform $λ$-property.

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arXiv2019

On the geometry of higher order Schreier spaces

For each countable ordinal $α$ let $\mathcal{S}_α$ be the Schreier set of order $α$ and $X_{\mathcal{S}_α}$ be the corresponding Schreier space of order $α$. In this paper we prove several new properties of these spaces. 1) If $α$ is non-zero then $X_{\mathcal{S}_α}$ possesses the $λ$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $α$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}_{\mathcal{S}_α}$ possesses the uniform $λ$-property of R. Aron and R. Lohman. 3) For each countable ordinal $α$ the space $X^*_{\mathcal{S}_α}$ has the $λ$-property. 4) For $n\in \mathbb{N}$, if $U:X_{\mathcal{S}_n}\to X_{\mathcal{S}_n}$ is an onto linear isometry then $Ue_i = \pm e_i$ for each $i \in \mathbb{N}$. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for non-zero $α$, $X_{\mathcal{S}_α}$ is $(V)$-polyhedral and has the $λ$-property implies that each $X_{\mathcal{S}_α}$ is an example of space solving a problem of J. Lindenstrauss from 1966. The first example of such a space was given by C. De Bernardi in 2017 using a renorming of $c_0$.

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