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Carlo Sanna

Publications and source records attributed to Carlo Sanna.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the l.c.m. of shifted Lucas numbers

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1 + s_1, L_2 + s_2, \dots, L_n + s_n)$ as $n \to +\infty$, for $(s_n)_{n \geq 1}$ a periodic sequence in $\{-1, +1\}$. We also carry out the same analysis for $(s_n)_{n \geq 1}$ a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$. These results are Lucas numbers-analogs of previous results obtained by the author for the sequence of Fibonacci numbers.

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arXiv2021

Membership in random ratio sets

Let $\mathcal{A}$ be a random set constructed by picking independently each element of $\{1, \dots, n\}$ with probability $α\in (0, 1)$. We give a formula for the probability that a rational number $q$ belong to the random ratio set $\mathcal{A} /\! \mathcal{A} := \{a / b : a,b \in \mathcal{A}\}$. This generalizes a previous result of Cilleruelo and Guijarro-Ordóñez. Moreover, we make some considerations about formulas for the probability of the event $\bigvee_{i=1}^k\!\big(q_i \in \mathcal{A} /\! \mathcal{A}\big)$, where $q_1, \dots, q_k$ are rational numbers, showing that they are related to the study of the connected components of certain graphs. In particular, we give formulas for the probability that $q^e \in \mathcal{A} /\! \mathcal{A}$ for some $e \in \mathcal{E}$, where $\mathcal{E}$ is a finite or cofinite set of positive integers with $1 \in \mathcal{E}$.

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arXiv2021

On the least common multiple of shifted powers

Let $a \geq 2$ be an integer. We prove that for every periodic sequence $(s_n)_{n \geq 1}$ in $\{-1, +1\}$ there exists an effectively computable rational number $C_\mathbf{s} > 0$ such that \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim C_\mathbf{s} \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*} as $n \to +\infty$, where $\operatorname{lcm}$ denotes the least common multiple. Furthermore, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$ then \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim 6 \operatorname{Li}_2\!\big(\tfrac1{2}\big) \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*} with probability $1 - o(1)$, as $n \to +\infty$, where $\operatorname{Li}_2$ is the dilogarithm function.

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arXiv2021

Additive bases and Niven numbers

Let $g \geq 2$ be an integer. A natural number is said to be a base-$g$ Niven number if it is divisible by the sum of its base-$g$ digits. Assuming Hooley's Riemann Hypothesis, we prove that the set of base-$g$ Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-$g$ Niven numbers.

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arXiv2020

On the least common multiple of random $q$-integers

For every positive integer $n$ and for every $α\in [0, 1]$, let $\mathcal{B}(n, α)$ denote the probabilistic model in which a random set $\mathcal{A} \subseteq \{1, \dots, n\}$ is constructed by picking independently each element of $\{1, \dots, n\}$ with probability $α$. Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $\mathcal{A}$. Let $q$ be an indeterminate and let $[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in \mathbb{Z}[q]$ be the $q$-analog of the positive integer $k$. We determine the expected value and the variance of $X := \operatorname{deg} \operatorname{lcm}\!\big([\mathcal{A}]_q\big)$, where $[\mathcal{A}]_q := \big\{[k]_q : k \in \mathcal{A}\big\}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.

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arXiv2020

On the divisibility of the rank of appearance of a Lucas sequence

Let $U = (U_n)_{n \geq 0}$ be a Lucas sequence and, for every prime number $p$, let $ρ_U(p)$ be the rank of appearance of $p$ in $U$, that is, the smallest positive integer $k$ such that $p$ divides $U_k$, whenever it exists. Furthermore, let $d$ be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes $p \leq x$ such that $d$ divides $ρ_U(p)$, as $x \to +\infty$.

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arXiv2020

The Largest Entry in the Inverse of a Vandermonde Matrix

We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real $b > 1$, let $M_b(n)$ be the maximum of the absolute values of the entries of the inverse of the $n \times n$ matrix $[b^{i j}]_{0 \leq i, j < n}$. We prove that $\lim_{n \to +\infty} M_b(n)$ exists, and we provide some formulas for it.

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arXiv2020

On the l.c.m. of shifted Fibonacci numbers

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log α}{π^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $α:= \big(1 + \sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\mathbf{s} = (s_n)_{n \geq 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}} > 0$ such that \begin{equation*} \log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n) \sim \frac{3 \log α}{π^2} \cdot C_\mathbf{s} \cdot n^2 , \quad \text{as } n \to +\infty . \end{equation*} Moreover, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \mathbb{E}\big[\log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n)\big] \sim \frac{3 \log α}{π^2} \cdot \frac{15 \operatorname{Li}_2(1 / 16)}{2} \cdot n^2 , \quad \text{as } n \to +\infty , \end{equation*} where $\operatorname{Li}_2$ is the dilogarithm function.

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arXiv2020

A note on the natural density of product sets

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $<\varepsilon$. This answers two questions of Hegyvári, Hennecart and Pach.

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arXiv2020

Practical central binomial coefficients

A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We improve this result by reducing the number of exceptions to $\exp\!\big(C (\log x)^{4/5} \log \log x\big)$, where $C > 0$ is a constant.

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arXiv2020

Greedy approximations by signed harmonic sums and the Thue--Morse sequence

Given a real number $τ$, we study the approximation of $τ$ by signed harmonic sums $σ_N(τ) := \sum_{n \leq N}{s_n(τ)}/n$, where the sequence of signs $(s_N(τ))_{N \in\mathbb{N}}$ is defined "greedily" by setting $s_{N+1}(τ) := +1$ if $σ_N(τ) \leq τ$, and $s_{N+1}(τ) := -1$ otherwise. Precisely, we compute the limit points and the decay rate of the sequence $(σ_N(τ)-τ)_{N \in \mathbb{N}}$. Moreover, we give an accurate description of the behavior of the sequence of signs $(s_N(τ))_{N\in\mathbb{N}}$, highlighting a surprising connection with the Thue--Morse sequence.

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arXiv2020

On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.

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arXiv2019

A note on product sets of random sets

Given two sets of positive integers $A$ and $B$, let $AB := \{ab : a \in A,\, b \in B\}$ be their product set and put $A^k := A \cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every $α\in [0,1]$, let $\mathcal{B}(n, α)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $α$. We prove that if $A_1, \dots, A_s$ are random sets in $\mathcal{B}(n_1, α_1), \dots, \mathcal{B}(n_s, α_s)$, respectively, $k_1, \dots, k_s$ are fixed positive integers, $α_i n_i \to +\infty$, and $1/α_i$ does not grow too fast in terms of a product of $\log n_j$; then $|A_1^{k_1} \cdots A_s^{k_s}| \sim \frac{|A_1|^{k_1}}{k_1!}\cdots\frac{|A_s|^{k_s}}{k_s!}$ with probability $1 - o(1)$. This is a generalization of a result of Cilleruelo, Ramana, and Ramaré, who considered the case $s = 1$ and $k_1 = 2$.

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arXiv2019

On the l.c.m. of random terms of binary recurrence sequences

For every positive integer $n$ and every $δ\in [0,1]$, let $B(n, δ)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $δ$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $α$ and $β$, with $|α| \geq |β|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $α/ β$ is not a root of unity. We prove that, as $δn / \log n \to +\infty$, for every $A$ in $B(n, δ)$ we have $$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{δ\operatorname{Li}_2(1 - δ)}{1 - δ} \cdot \frac{3\log\!\big|α/ \!\sqrt{(a_1^2, a_2)}\big|}{π^2} \cdot n^2 $$ with probability $1 - o(1)$, where $\operatorname{lcm}$ denotes the lowest common multiple, $\operatorname{Li}_2$ is the dilogarithm, and the factor involving $δ$ is meant to be equal to $1$ when $δ= 1$. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case $δ= 1$, and is motivated by an asymptotic formula for $\operatorname{lcm}(A)$ due to Cilleruelo, Rué, Šarka, and Zumalacárregui.

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arXiv2019

Directions sets: A generalization of ratio sets

For every integer $k \geq 2$ and every $A \subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^k\}$ and $D^{\underline{k}}(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^{\underline{k}}\}$, where $\|\cdot\|$ is the Euclidean norm and $A^{\underline{k}} := \{{\bf a} \in A^k : a_i \neq a_j \text{ for all } i \neq j\}$. Via an appropriate homeomorphism, $D^k(A)$ is a generalization of the \emph{ratio set} $R(A) := \{a / b : a,b \in A\}$, which has been studied by many authors. We study $D^k(A)$ and $D^{\underline{k}}(A)$ as subspaces of $S^{k-1} := \{{\bf x} \in [0,1]^k : \|{\bf x}\| = 1\}$. In~particular, generalizing a result of Bukor and Tóth, we provide a characterization of the sets $X \subseteq S^{k-1}$ such that there exists $A \subseteq \mathbb{N}$ satisfying $D^{\underline{k}}(A)^\prime = X$, where $Y^\prime$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^k(A)$ to be dense in $S^{k-1}$. We conclude leaving some questions for further research.

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arXiv2019

Counting Egyptian fractions

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.

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arXiv2019

Practical numbers among the binomial coefficients

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting $f(n)$ denote the number of binomial coefficients $\binom{n}{k}$, with $0 \leq k \leq n$, that are not practical numbers, we show that \begin{equation*} f(n) < n^{1 - (\log 2 - δ)/\log \log n} \end{equation*} for all integers $n \in [3, x]$, but at most $O_γ(x^{1 - (δ- γ) / \log \log x})$ exceptions, for all $x \geq 3$ and $0 < γ< δ< \log 2$. Furthermore, we prove that the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions. We also pose some questions on this topic.

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