On the distribution of sparse sequences in prime fields and applications

Víctor Cuauhtemoc García

Displaying similar documents to “On the distribution of sparse sequences in prime fields and applications”

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

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We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

On the sum of dilations of a set

Antal Balog, George Shakan (2014)

Acta Arithmetica

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We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has ${| p \cdot A + q \cdot A | \geq (p + q) | A | - (p q)}^{(p + q - 3) (p + q) + 1}$ .

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

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We say a sequence $= {(s ₙ)}_{n \geq 0}$ is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, $_{a} = {(u ₙ)}_{n \geq 0}$ , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences $_{a} \pm k$ are simultaneously primefree. This...

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël Rivat (2015)

Journal of the European Mathematical Society

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For a large class of digital functions $f$ , we estimate the sums $\sum_{n \leq x} Λ (n) f (n)$ (and $\sum_{n \leq x} μ (n) f (n)$ , where $Λ$ denotes the von Mangoldt function (and $μ$ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

On the distribution of prime numbers which are of the form $x^{2} + y^{2} + 1$

Yoichi Motohashi (1970)

Acta Arithmetica

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On square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n + 1} = P U ₙ + Q U_{n - 1}$ , $V_{n + 1} = P V ₙ + Q V_{n - 1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

Equicontinuity and Convergent Sequences in the Spaces $_{C}^{'}$ and $_{M}$

Jan Kisyński (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Characterizations of equicontinuity and convergent sequences are given for the space $_{C}^{'} (ℝ ⁿ)$ of rapidly decreasing distributions and the space $_{M} (ℝ ⁿ)$ of slowly increasing infinitely differentiable functions.

On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let $p_{1}, p_{2}, \dots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k \geq 5$ , then $(p_{k + 1} - 1)! ∣ (\frac{1}{2} (p_{k + 1} - 1))! p_{k}!,$ which improves a previous result of the second author.

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\operatorname{sgn}x\geq 0$ for $|x|>R$ .

Ján Andres (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi, John R. Graef (2023)

Mathematica Bohemica

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We study the existence of positive solutions to the fourth-order two-point boundary value problem $\{\begin{matrix} u^{''''} (t) + f (t, u (t)) = 0, & 0 < t < 1, \\ u^{'} (0) = u^{'} (1) = u^{''} (0) = 0, & u (0) = α [u], \end{matrix}$ where $α [u] = \int_{0}^{1} u (t) d A (t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in 𝒞 ([0, 1] \times ℝ_{+}, ℝ_{+})$ . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

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Integer sequences of the form $⌊ n^{c} ⌋$ , where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference $\sum_{n \leq x} φ (⌊ n^{c} ⌋) - 1 / c \sum_{n \leq x^{c}} φ (n) n^{1 / c - 1}$ , which measures the deviation of the mean value of φ on the subsequence $⌊ n^{c} ⌋$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $⌊ n^{c} ⌋$ attains both of its values with...

$E_{1}$ -degeneration and $d^{'} d^{''}$ -lemma

Tai-Wei Chen, Chung-I Ho, Jyh-Haur Teh (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a double complex $(A, d^{'}, d^{''})$ , we show that if it satisfies the $d^{'} d^{''}$ -lemma and the spectral sequence ${E_{r}^{p, q}}$ induced by $A$ does not degenerate at $E_{0}$ , then it degenerates at $E_{1}$ . We apply this result to prove the degeneration at $E_{1}$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{'} d^{''}$ -lemma.

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

Boundedness criteria for a class of second order nonlinear differential equations with delay

Daniel O. Adams, Mathew Omonigho Omeike, Idowu A. Osinuga, Biodun S. Badmus (2023)

Mathematica Bohemica

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We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $a (t) x^{''} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'})$ and ${(a (t) x^{'})}^{'} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'}),$ where $a$ , $b$ , $c$ , $g$ , $h$ , $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr Mykhaylyuk, Anatolij Plichko (2015)

Colloquium Mathematicae

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We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_{y} (x) : = f_{y}^{'} (x, y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f_{y x}^{''}$ exists. We construct a separately twice differentiable function whose partial derivative $f_{x}^{'}$ is discontinuous with respect to the second variable on a...

On the non-convergence of successive approximations in the Darboux problem for the equation $z_{x y}^{''} = f (x, y, z)$

M. Kisielewicz (1977)

Colloquium Mathematicae

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Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

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Letting f(n) = A log n + t(n), where t(n) is a small additive function and A a positive constant, we obtain estimates for the quantities $\sum_{x \leq n \leq x + H} 1 / f (Q (n))$ and $\sum_{x \leq p \leq x + H} 1 / f (Q (p))$ , where H = H(x) satisfies certain growth conditions, p runs over prime numbers and Q is a polynomial with integer coefficients, whose leading coefficient is positive, and with all its roots simple.

Elementary operators on Banach algebras and Fourier transform

Miloš Arsenović, Dragoljub Kečkić (2006)

Studia Mathematica

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We consider elementary operators $x \mapsto \sum_{j = 1}^{n} a_{j} x b_{j}$ , acting on a unital Banach algebra, where $a_{j}$ and $b_{j}$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_{j}$ and $b_{j}$ , i.e. $a_{j} = a_{j}^{'} + i a_{j}^{''}$ ( $b_{j} = b_{j}^{'} + i b_{j}^{''}$ ), where all $a_{j}^{'}$ and $a_{j}^{''}$ ( $b_{j}^{'}$ and $b_{j}^{''}$ ) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

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Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)} (ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)} (ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Radu Ignat, Felix Otto (2008)

Journal of the European Mathematical Society

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In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^{1}$ -valued maps $m^{'}$ (the magnetization) of two variables $x^{'}$ : $E_{ε} (m^{'}) = ε \int {| \nabla^{'} \cdot m^{'} |}^{2} d x^{'} + \frac{1}{2} \int {|| \nabla^{'} |^{- 1 / 2} \nabla^{'} \cdot m^{'}|}^{2} d x^{'}$ . We are interested in the behavior of minimizers as $ε \to 0$ . They are expected to be $S^{1}$ -valued maps $m^{'}$ of vanishing distributional divergence $\nabla^{'} \cdot m^{'} = 0$ , so that appropriate boundary conditions enforce line discontinuities. For finite $ε > 0$ , these line discontinuities are approximated by smooth transition layers, the so-called Néel...