Piatetski-Shapiro Sequences Research Articles

Partitions into Segal–Piatetski–Shapiro sequences

Partitions into Segal–Piatetski–Shapiro sequences

A generalization of Piatetski–Shapiro sequences (II)

A generalization of Piatetski–Shapiro sequences (II)

Bracket words along Hardy field sequences

AbstractWe study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $ . We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

Exponential sums with Piatetski-Shapiro sequences and their application to modular hyperbolas

The Piatetski-Shapiro sequence associated with c>1 and c∉N is given by Nc=(⌊nc⌋)n∈N. Let p be an odd prime and let g(x),h(x)∈Fp[x] be coprime polynomials. We estimate exponential sums with Piatetski-Shapiro sequences of the shape∑m≤Mm∈Nc,(h(m),p)=1ep(g(m)h(m)), where ep(y)=exp⁡(2πiy/p). In particular, when g(x)/h(x)=rx+sx¯ and 1<c<6/5, our results improve the bound for Kloosterman sums with Piatetski-Shapiro sequences which was given by Shparlinski and Technau. Furthermore, bounds for max⁡{m,m˜} subject to m,m˜∈Z∩[1,p), z indivisible by p, g(m)m˜≡z(modp) and m belonging to some fixed Piatetski-Shapiro sequence are also obtained.

$ k $th powers in a generalization of Piatetski-Shapiro sequences

&lt;abstract&gt;&lt;p&gt;The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c &amp;gt; 1 $, and $ \beta $ are real numbers.&lt;/p&gt; &lt;p&gt;The focus of the study is on solving equations of the form $ \left\lfloor \alpha n^c +\beta\right\rfloor = s m^k $, where $ m $ and $ n $ are positive integers, $ 1 \leq n \leq N $, and $ s $ is an integer. Bounds for the solutions are obtained for different values of the exponent $ k $, and an average bound is derived over $ k $-free numbers $ s $ in a given interval.&lt;/p&gt;&lt;/abstract&gt;

On a question of Luca and Schinzel over Segal–Piatetski-Shapiro sequences

We extend to Segal–Piatetski-Shapiro sequences previous results on the Luca–Schinzel question. Namely, we prove that for any real c larger than 1, the sequence \((\sum _{m\le n} \varphi (\lfloor m^c \rfloor ) /\lfloor m^c \rfloor )_{n\ge 1}\) is dense modulo 1, where \(\varphi \) denotes Euler’s totient function. The main part of the proof consists in showing that when R is a large integer, the sequence of the residues of \(\lfloor m^c \rfloor \) modulo R contains any block of consecutive residues of a given length.

Multiplicative functions of special type on Piatetski-Shapiro sequences

Abstract Let f k ( n ) := ∑ d k ∣ n Φ ( d ) , $$ f_k(n):=\sum\limits_{d^k \mid n} \Phi(d), $$ where Φ(d) is a multiplicative function, Φ(d) = O(dε ). We study asymptotic behaviour of the sum T f k c ( N ) := ∑ n ≤ N f k n c , 1 &lt; c &lt; 2 , $$ T_{f_k}^c(N):=\sum\limits_{n \leq N} f_k\left(\left\lfloor n^c\right\rfloor\right), \quad 1&lt;c&lt;2, $$ and establish some asymptotic formulas for T f k c ( N ) $ T_{f_k}^c(N) $ under assumptions on fk .

Average value of some certain types of arithmetic functions with Piatetski-Shapiro sequences

In this paper, we study asymptotic behaviour of the sum $\sum_{n\leq N}{f}\Big(\lfloor n^c \rfloor\Big),$ where $f(n)=\sum_{d^2\mid n}g(d)$ under three different types of assumptions on $g$ and $1&amp; &lt; c &lt; 2$.

Representations of large integers as the sum of fractional powers of primes and squares

Conjecture H of Hardy and Littlewood says that every large integer is either a square or the sum of a prime and a square. Despite progress showing that this is true for almost all large integers, Conjecture H remains unproven. In this paper, Conjecture H is approximated using Piatetski–Shapiro sequences by showing that all large integers can be written in the form [Formula: see text] for any fixed [Formula: see text]. The existing results for the conjecture are also extended by showing that almost all large integers can be written in the form [Formula: see text] for any fixed [Formula: see text].

Kth powers in Piatetski-Shapiro sequences

The Piatetski-Shapiro sequences are sequences of the form [Formula: see text] with [Formula: see text] and [Formula: see text]. We investigate the distribution of [Formula: see text]th powers in Piatetski-Shapiro sequences. Moreover, we study the solution of a more general equation [Formula: see text] with [Formula: see text] for an integer [Formula: see text]. We obtain bounds for different values of [Formula: see text] and an average bound over [Formula: see text]-free numbers [Formula: see text] in an interval. In the end, we also study the solution of the equation [Formula: see text] for which [Formula: see text] is a prime number.

Additive energy and a large sieve inequality for sparse sequences

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials f ( X ) = X k $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K. Halupczok (2012, 2015, 2018) and M. Munsch (2020). We also consider moduli defined by polynomials f ( X ) ∈ Z [ X ] $f(X) \in \mathbb {Z}[X]$ , Piatetski–Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri–Vinogradov theorem with Piatetski–Shapiro moduli improving the level of distribution of R. C. Baker (2014).

A Generalization of Piatetski–Shapiro Sequences

We consider a generalization of Piatetski–Shapiro sequences in the sense of Beatty sequences, which is of the form $(\lfloor \alpha n^c + \beta \rfloor)_{n=1}^{\infty}$ with real numbers $\alpha \geq 1$, $c \gt 1$ and $\beta$. We show there are infinitely many primes in the generalized Piatetski–Shapiro sequence with $c \in (1,14/13)$. Moreover, we prove there are infinitely many Carmichael numbers composed entirely of the primes from the generalized Piatetski–Shapiro sequences with $c \in (1,64/63)$.

Linear equations with two variables in Piatetski-Shapiro sequences

For every non-integral $\alpha \gt 1$, the sequence of the integer parts of $n^{\alpha }$ $(n=1,2,\ldots )$ is called the Piatetski-Shapiro sequence with exponent $\alpha $ and let $\mathrm {PS}(\alpha )$ denote the set of all terms of this sequence. For

On squares in Piatetski–Shapiro sequences

On squares in Piatetski–Shapiro sequences

Distributions of finite sequences represented by polynomials in Piatetski-Shapiro sequences

By using the work of Frantzikinakis and Wierdl, we can see that for all d∈N, α∈(d,d+1), and integers k≥d+2 and r≥1, there exist infinitely many n∈N such that the sequence (⌊(n+rj)α⌋)j=0k−1 is represented as ⌊(n+rj)α⌋=p(j), j=0,1,…,k−1, by using some polynomial p(x)∈Q[x] of degree at most d. In particular, the above sequence is an arithmetic progression when d=1. In this paper, we show the asymptotic density of such numbers n as above. When d=1, the asymptotic density is equal to 1/(k−1). Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields.

Linear Diophantine equations in Piatetski-Shapiro sequences

A Piatetski-Shapiro sequence with exponent $\alpha$ is a sequence of integer parts of $n^\alpha$ $(n = 1,2,\ldots)$ with a non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set of those terms. In this article, we study the set of $\alpha$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\alpha > 2$ such that $\mathrm{PS}(\alpha)$ contains infinitely many arithmetic progressions of length $3$.

Almost primes in Piatetski-Shapiro sequences

<abstract> The Piatetski-Shapiro sequences are sequences of the form $ (\left\lfloor {{n^c}} \right\rfloor)_{n = 1}^\infty $ for $ c &gt; 1 $ and $ c \not\in \mathbb{N} $. It is conjectured that there are infinitely many primes in Piatetski-Shapiro sequences for $ c \in (1, 2) $. For every $ R \ge 1 $, we say that a natural number is an $ R $-almost prime if it has at most $ R $ prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many $ R $-almost primes in Piatetski-Shapiro sequences if $ c \in (1, c_R) $ and $ c_R $ is an explicit constant depending on $ R $. </abstract>

Square-full numbers in Piatetski–Shapiro sequences

A positive integer n is called square-full if for every prime $$p\vert n$$ , also $$p^2\vert n$$ . Piatetski–Shapiro sequences (PS-sequences) are defined by $$\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \quad (c>1, c\notin {\mathbb {N}}), \end{aligned}$$ where $$\lfloor z\rfloor $$ is the integer part of a real z. In this paper we investigate the distribution of square-full numbers in Piatetski–Shapiro sequences. Resume Un entier positif n est appele un nombre puissant si pour chaque nombre premier p qui divise n, on a $$p^2\mid n$$ . Les sequences de Piatetski–Shapiro (PS-sequences) sont definies par $$\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \ \ \ \ (c>1, c\notin {\mathbb {N}}), \end{aligned}$$ ou $$\lfloor z\rfloor $$ est la partie entie entiere d’un reel z. Dans cet article, nous etudions la distribution des nombres puissants dans des sequences de Piatetski–Shapiro.

A perturbed Khinchin-type theorem and solutions to linear equations in Piatetski-Shapiro sequences

Our main result concerns a perturbation of a classic theorem of Khinchin in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(\psi _n)_{n \in \mathbb N }$ and differentiable functions $(\varphi _n: J \to \ma

Kloosterman sums with twice-differentiable functions

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \floor{f(n)}+ y \floor{f(n)}^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\floor{f(n)}^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\floor{f(n)}^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \floor{t^c}$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.