arXiv subjects

Deepa Antony

Publications and source records attributed to Deepa Antony.

7 recordsLinked to original sources

arXiv2024

$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017, Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha$, where $\alpha$ is a Pisot number and if $p$ is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb{Q}_p$, then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb{Q}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb{Q}$.

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arXiv2024

On the $p$-adic valuation of third order linear recurrence sequences

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a generalisation of the conjecture of Marques and Lengyel for third order linear recurrence sequences. Suppose that $(x_n)$ is a third order linear recurrence sequence whose characteristic polynomial has a root $\gamma$ such that $|\gamma|>1$. We show that if there exists a prime $p$ for which the conjecture holds for $(x_n)$, then the solution set of the Diophantine equation given by $x_n=m!$ in positive integers $n,m$ is finite. We also show that the solutions can be effectively computed when the form of the conjecture is explicitly known.

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arXiv2023

On $p$-adic denseness of quotients of values of integral forms

Given $A\subseteq \Z$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained by an integral form. For a given form, we investigate whether this happens for all but finitely many $p$. We consider the more general question when forms have coefficients in Dedekind domains, and, under certain conditions, we prove that the analogous statement holds. However, we also give examples of integral forms for which the answer is negative, and it is crucial that the degree of such forms is composite. We conjecture that we cannot find such examples with forms of prime degree having sufficiently many variables, which is indeed the case when the degree is 2, 3, or 5. Our innovation is to consider the problem in terms of varieties defined by forms, thereby using the tools from algebraic geometry, making their first appearance in this setting. We construct integral forms such that the ratio set of its values is dense in at least one $\Q_p$ but only in finitely many of them.

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arXiv2022

$p$-Adic quotient sets: linear recurrence sequences

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linear recurrence $(x_n)_{n\geq0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\dots=x_{k-2}=0,x_{k-1}=1$, where $a_1,\dots,a_k\in \mathbb{Z}$ and $a_k=1$. We show that given a prime $p$, there exist infinitely many recurrence sequences of order $k\geq 2$ so that their quotient sets are not dense in $\mathbb{Q}_p$. We also study the quotient sets of linear recurrence sequences with coefficients in some arithmetic and geometric progressions.

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arXiv2022

On denseness of certain direction and generalized direction sets

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets of positive integers so that the corresponding $k$-generalized direction sets are dense in $\mathcal{S}^{k - 1}$. We also consider the denseness properties of certain direction sets and give a partial answer to a question posed by Leonetti and Sanna. Finally we consider a similar question in the framework of an algebraic number field.

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arXiv2022

$p$-Adic quotient sets: diagonal forms

For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic denseness of quotients of nonzero values attained by diagonal forms of degree $n\geq 3$, where $\gcd(n,p(p-1))=1$.

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arXiv2021

$p$-adic quotient sets: Cubic forms

For $A\subseteq \{1, 2, \ldots\}$, we consider $R(A)=\{a/b: a, b\in A\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. We also prove that if $A$ is the set of nonzero values assumed by a non-degenerate, integral and primitive cubic form with more than 9 variables, then $R(A)$ is dense in $\mathbb{Q}_p$.

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