For every non-integral $α>1$, the sequence of the integer parts of $n^α$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $α$, and let $\mathrm{PS}(α)$ denote the set of all those terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is solvable in $X$ if the equation has infinitely many solutions of distinct pairs $(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\neq 1$ and $0\leq b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<α<2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$. Further, we investigate the set of $α\in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$ where $2< s <t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.
A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.
We can give a slight improvement for a result related to the distribution of squares in Piatetski-Shapiro sequences of Liu-Shparlinski-Zhang \cite{lsz} by involving three different ideas.
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
In this paper, we exhibit an asymptotic formula for the number of representations of a large integer as a sum of a fixed power of Piatetski-Shapiro primes, thereby establishing a variant of Waring-Goldbach problem with primes from a sparse sequence.
In this paper, we study the hybrid problem of Hua's theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case $k=3$, which deepen the classical result of Hua.
We show that the Möbius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the squares $t(n^2)$. These are both examples of Möbius orthogonal sequences with maximum entropy.
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/γ}]$ with $\frac{85}{86}<γ<1$. Moreover, we use this result to prove that, for $0.9989445<γ<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.
Let $α,β$ be real numbers such that $α>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.
Let $c > 1$ and $0 < γ< 1$ be real, with $c \notin \mathbb N$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2, \dots, p_s$ of index $γ$---that is, primes of the form $p = \lfloor m^{1/γ} \rfloor$ for some $m \in \mathbb N$.
Suppose that $α,β\in\mathbb{R}$. Let $α\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which is defined by $(\lfloorαn^c+β\rfloor)_{n=1}^\infty$. Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from the generalized Piatetski--Shapiro sequences with $c\in(1,\frac{19137}{18746})$. The two theorems constitute improvements upon previous results by Guo and Qi.
In this paper, we prove that for any fixed $205/243<γ\leqslant1$, every sufficiently large $N$ satisfying $N\equiv 5 \pmod {24}$ can be represented as five squares of primes with one prime in $\mathcal{P}_γ$, which improves the previous result of Zhang and Zhai.
Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).
We study the distribution of squares in a Piatetski-Shapiro sequence $\left(\lfloor n^c\rfloor\right)_{n\in\mathbb N}$ with $c>1$ and $c\not\in\mathbb N$. We also study more general equations $\lfloor{n^c}\rfloor = sm^2$, $n,m\in \mathbb N$, $1\le n \le N$ for an integer $s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.
Suppose that $α_1, α_2,β_1, β_2 \in\mathbb{R}$. Let $α_1, α_2 > 1$ be irrational and of finite type such that $1, α_1^{-1}, α_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range $1 < c < 12/11$. In this paper, it is proved that there exist infinitely many primes in the intersection of Beatty sequences $\mathcal{B}_{α_1,β_1} = \lfloorα_1 n + β_1\rfloor, \mathcal{B}_{α_2, β_2} = \lfloorα_2 n + β_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the inter…
Let $λ(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $λ(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq N$.
The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$ satisfies a prime number theorem of the form $\sum_{n\le x} Λ(n)a(\lfloor n^c \rfloor) \sim C\, x$, and, furthermore, that it is deterministic for $c \in \mathbb R\setminus \mathbb Z$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor \bmod m$ has polynomial subword complexity.
We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ 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 blocks of consecutive values which are in an arithmetic progression.
Let $m\geq 3$. Suppose that $$ 1-2^{-2^{m^24^m}}<γ<1. $$ Then the set $$ \{p\text{ prime}:\, p=[n^{\frac1γ}]\text{ for some }n\in{\mathbb N}\} $$ contains infinitely many non-trivial $m$-term arithmetic progressions.
We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $θ(c)$ with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{θ(c)-\eps}$ infinitely often. For $c\in(1,\tfrac{149}{87})$ we show that the counting function of natural numbers $n\le x$ for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For $c\in(1,\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.
In this paper, we prove a quantitative version of the statement that every nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by S. Wright in certain aspects.
Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $η$ is real, the constants $λ_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,\,2,\,3$.
We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.
In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $\mathrm{SO}(n,1)$ constructed by Gromov and Piatetski-Shapiro can be embedded into $\mathrm{SL}_{n+1}(\mathbb{R})$ so that their images are thin subgroups
Denote by $\text{PS}(α)$ the image of the Piatetski-Shapiro sequence $n \mapsto \lfloor n^α \rfloor$ where $α> 1$ is non-integral and $\lfloor x \rfloor$ is the integer part of $x \in \mathbb{R}$. We partially answer the question of which bivariate linear equations have infinitely many solutions in $\text{PS}(α)$: if $a, b \in \mathbb{R}$ are such that the equation $y=ax+b$ has infinitely many solutions in the positive integers, then for Lebesgue-a.e. $α> 1$, it has infinitely many or at most finitely many solutions in $\text{PS}(α)$ according as $α< 2$ (and $0 \leq b < a$) or $α> 2$ (and $(a,…
We study Piatetski-Shapiro sequences $(\lfloor n^c\rfloor)_n$ modulo m, for non-integer $c >1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in\{0,1\}^k$ of length $k < c + 1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi-Kátai criterion, we prove that t…
Let $κ$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{κ, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^κ\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{κ, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.
Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results…
Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfl…
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(ψ_n)_{n \in \mathbb{N}}$ and differentiable functions $(φ_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so that for Lebesgue-a.e. $θ\in J$, the inequality $\| nθ+ φ_n(θ) \| \leq ψ_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $|φ_n(θ)|$ is allowed to exceed $ψ_n$, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main resul…
By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $α\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^α}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^α}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, by using some polynomial $p(x)\in\mathbb{Q}[x]$ 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 $1/(k-1)$…
Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences
School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）；Research Center for Number Theory and Its Applications, Northwest University（西北大学数论及其应用研究中心）
