arXiv subjects

Vitaly Bergelson

Publications and source records attributed to Vitaly Bergelson.

At least 37 records · Page 2Linked to original sources

arXiv2023

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. By building on the ideas behind our ergodic results, we recast Sarnak's Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases.

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arXiv2023

Joint normality of representations of numbers: an ergodic approach

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with {\em pointwise joint ergodicity}: Let $T_b:[0,1] \rightarrow [0,1]$ be the times $b$ map defined by $T_b x = bx \, \bmod \, 1 $ and let $T_G:[0,1] \rightarrow [0,1]$ be the Gauss map defined by $T_G(x) = \{\frac{1}{x}\}$ for $x \ne 0$ and $T_G (0) =0.$ (Here $\{ \cdot \}$ denotes the fractional part.) For any $f, g \in L^{\infty} (λ)$, \[ \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{n=0}^{N-1} f(T_b^{n}x) \, g(T_G^n x) = \int f \, d λ\cdot \int g \, d μ_G \quad \text{for almost every } x \in [0,1], \] where $λ$ is the Lebesgue measure on $[0,1]$ and $μ_G$ is the Gauss measure on $[0,1]$ given by $μ_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$. We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of number-theoretical maps of the interval and derive the corresponding corollaries pertaining to joint normality. We also establish the equivalence of various forms of normality and joint normality for representations of numbers, hereby providing a general framework for classical normality results.

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arXiv2023

Multiple recurrence and popular differences for polynomial patterns in rings of integers

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If $K$ is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = δ> 0$, we show, inter alia: 1. If $p(x) \in K[x]$ is an intersective $\mathcal{O}_K$-valued polynomial and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$, the set of $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n)\} \subseteq E \} \right) > δ^3 - \varepsilon. \] is syndetic. Moreover, if $\frac{s}{r} \in \mathbb{Q}$, then there are syndetically many $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \} \right) > δ^4 - \varepsilon. \] 2. If $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family of linearly independent $\mathcal{O}_K$-valued polynomials, then the set of $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \} \right)> δ^{k+1} - \varepsilon \] is syndetic. These two results generalize and extend previous work of Frantzikinakis and Kra on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory, which require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalization of Weyl's equidistribution theorem.

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arXiv2023

The Furstenberg-Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $μ(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings $\mathbb{Z}/N_m\mathbb{Z}$, $m \in \mathbb{N},$ is asymptotically totally ergodic if and only if $\mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $δ\in (0,1]$ and a (not necessarily intersective) polynomial $Q \in \mathbb{Q}[n]$ such that $Q(\mathbb{Z}) \subseteq \mathbb{Z}$, and write $S = \{ Q(n) : n \in \mathbb{Z}/N\mathbb{Z}\}$. For any integer $N > 1$ with $\mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $\mathbb{Z}/N\mathbb{Z}$ such that $|A||B| \geq δN^2$, then $\mathbb{Z}/N\mathbb{Z} = A + B + S$.

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arXiv2023

Polynomial patterns in subsets of large finite fields of low characteristic

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemerédi theorem for certain families of polynomials. Namely, we show that if $P_1, \dots, P_m \in (\mathbb{F}_p[t])[y]$ satisfy an equidistribution condition, which is a natural variant of the independence condition in (Peluse, 2019) for our context, then there exists $γ> 0$ such that for any $q = p^k$ and any $A_0, A_1, \dots, A_m \subseteq \mathbb{F}_q$, \begin{align*} \left| \left\{ (x,y) \in \mathbb{F}_q^2 : x \in A_0, x + P_1(y) \in A_1, \dots, x + P_m(y) \in A_m \right\} \right| = q^{-(m-1)} \prod_{i=0}^m{|A_i|} + O_{q \to \infty; P_1, \dots, P_m} \left( |A_0|^{1/2} q^{3/2 - γ} \right). \end{align*} In particular, if $A \subseteq \mathbb{F}_q$ contains no pattern $\{x, x + P_1(y), \dots, x + P_m(y)\}$ of cardinality $m+1$, then \begin{align*} |A| \ll_{P_1, \dots, P_m} q^{1 - γ/ \left( m + \frac{1}{2} \right)}. \end{align*}

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arXiv2022

Joint ergodicity of piecewise monotone interval maps

For $i = 0, 1, 2, \dots, k$, let $μ_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $λ$ and let $T_i:[0,1] \rightarrow [0,1]$ be $μ_i$-preserving ergodic transformations. We say that transformations $T_0, T_1, \dots, T_k$ are uniformly jointly ergodic with respect to $(λ; μ_0, μ_1, \dots, μ_k)$ if for any $f_0, f_1, \dots, f_k \in L^{\infty}$, \[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \, d μ_i \quad \text{ in } L^2(λ). \] We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let $T_G$ denote the Gauss map, $T_G(x) = \frac{1}{x} \, (\bmod \, 1)$, and, for $β>1$, let $T_β$ denote the $β$-transformation defined by $T_β x = βx \, (\bmod \,1)$. Let $T_0$ be an ergodic interval exchange transformation. Let $β_1 , \cdots , β_k$ be distinct real numbers with $β_i >1$ and assume that $\log β_i \ne \frac{π^2}{6 \log 2}$ for all $i = 1, 2, \dots, k$. Then for any $f_{0}, f_1, f_{2}, \dots, f_{k+1} \in L^{\infty} (λ)$, \begin{equation*} \begin{split} \lim\limits_{N -M \rightarrow \infty} \frac{1}{N -M } \sum\limits_{n=M}^{N-1} & f_{0} (T_0^n x) \cdot f_{1} (T_{β_1}^n x) \cdots f_{k} (T_{β_k}^n x) \cdot f_{k+1} (T_G^n x) &= \int f_{0} \, d λ\cdot \prod_{i=1}^k \int f_{i} \, d μ_{β_i} \cdot \int f_{k+1} \, d μ_G \quad \text{in } L^{2}(λ). \end{split} \end{equation*} We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

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arXiv2022

Discordant sets and ergodic Ramsey theory

We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemerédi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.

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arXiv2022

Khintchine-type recurrence for 3-point configurations

The goal of this paper is to generalize, refine, and improve results on large intersections. We show that if $G$ is a countable abelian group and $φ, ψ: G \to G$ are homomorphisms such that at least two of the three subgroups $φ(G)$, $ψ(G)$, and $(ψ-φ)(G)$ have finite index in $G$, then $\{φ, ψ\}$ has the \emph{large intersections property}. That is, for any ergodic measure preserving system $X=(X,\mathcal{X},μ,(T_g)_{g\in G})$, any $A\in\mathcal{X}$, and any $\varepsilon>0$, the set $$\{g\in G : μ(A\cap T_{φ(g)}^{-1} A\cap T_{ψ(g)}^{-1}A)>μ(A)^3-\varepsilon\}$$ is syndetic. Moreover, in the special case where $φ(g)=ag$ and $ψ(g)=bg$ for $a,b\in\mathbb{Z}$, we show that we only need one of the groups $aG$, $bG$, or $(b-a)G$ to be of finite index in $G$, and we show that the property fails in general if all three groups are of infinite index. One particularly interesting case is where $G=(\mathbb{Q}_{>0},\cdot)$ and $φ(g)=g$, $ψ(g)=g^2$, which leads to a multiplicative version for the large intersection result of Bergelson-Host-Kra. We also completely characterize the pairs of homomorphisms $φ,ψ$ that have the large intersections property when $G=\mathbb{Z}^2$. The proofs of our main results rely on analysis of the structure of the \emph{universal characteristic factor} for the multiple ergodic averages $$\frac{1}{|Φ_N|} \sum_{g\in Φ_N}T_{φ(g)}f_1\cdot T_{ψ(g)} f_2.$$ In the case where $G$ is finitely-generated, the characteristic factor for such averages is the \emph{Kronecker factor}. In this paper, we study actions of groups that are not necessarily finitely-generated, showing in particular that by passing to an extension of $X$, one can describe the characteristic factor in terms of the \emph{Conze--Lesigne factor} and the $σ$-algebras of $φ(G)$ and $ψ(G)$ invariant functions.

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arXiv2021

Multiple recurrence and large intersections for abelian group actions

The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $φ, ψ: G \to G$ are homomorphisms such that $φ(G)$, $ψ(G)$, and $(ψ- φ)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, μ, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$, the set $\{g \in G : μ(A \cap T_{φ(g)}^{-1}A \cap T_{ψ(g)}^{-1}A) > μ(A)^3 - \varepsilon\}$ is syndetic. (2) If $G$ is a countable abelian group and $r,s \in \mathbb{Z}$ are integers such that $rG$, $sG$, and $(r \pm s)G$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, μ, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$, the set $\{g \in G : μ(A \cap T_{rg}^{-1}A \cap T_{sg}^{-1}A \cap T_{(r+s)g}^{-1}A) > μ(A)^4 - \varepsilon\}$ is syndetic. In particular, these extend and generalize results of Bergelson, Host, and Kra concerning $\mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $\mathbb{F}_p^{\infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. We also discuss numerous examples shedding light on the necessity of the various hypotheses above. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper.

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arXiv2021

Strongly mixing systems are almost strongly mixing of all orders

We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP$^*$. While the main goal of this paper is to establish a $\textit{universal}$ property of strongly mixing actions of countable abelian groups, our results, when applied to $\mathbb Z$-actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for $\mathbb Z$-actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemerédi's theorem. We also demonstrate the versatility of $\mathcal R$-limits by obtaining new characterizations of higher order weak and mild mixing for actions of countable abelian groups.

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arXiv2020

Sums of Powers in Large Finite Fields: A Mix of Methods

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

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arXiv2020

An ergodic correspondence principle, invariant means and applications

A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with $d^*(E)>0$, where $d^*$ denotes the upper Banach density, then for any $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that $d^*\left(\bigcup_{i=1}^N(E-i)\right) > 1-\varepsilon$. Curiously, this result does not hold if one replaces the upper Banach density $d^*$ with the upper density $\bar{d}$. Originally proved combinatorially, Hindman's theorem allows for a quick and easy proof using an ergodic version of Furstenberg's correspondence principle. In this paper, we establish a variant of the ergodic Furstenberg's correspondence principle for general amenable (semi)-groups and obtain some new applications, which include a refinement and a generalization of Hindman's theorem and a characterization of countable amenable minimally almost periodic groups.

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arXiv2020

Uniqueness of a Furstenberg system

Given a countable amenable group $G$, a Følner sequence $(F_N) \subseteq G$, and a set $E \subseteq G$ with $\bar{d}_{(F_N)}(E)=\limsup_{N \to \infty} \frac{|E \cap F_N|}{|F_N|}>0$, Furstenberg's correspondence principle associates with the pair $(E,(F_N))$ a measure preserving system $(X,\mathcal{B},μ,(T_g)_{g \in G})$ and a set $A \in \mathcal{B}$ with $μ(A)=\bar{d}_{(F_N)}(E)$, in such a way that for all $r \in \mathbb{N}$ and all $g_1,\dots,g_r \in G$ one has $\bar{d}_{(F_N)}(g_1^{-1}E \cap \dots \cap g_r^{-1}E)\geqμ((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show that under some natural assumptions, the system $(X,\mathcal{B},μ,(T_g)_{g \in G})$ is unique up to a measurable isomorphism. We also establish variants of this uniqueness result for non-countable discrete amenable semigroups as well as for a generalized correspondence principle which deals with a finite family of bounded functions $f_1,\dots,f_{\ell}: G \rightarrow \mathbb{C}$.

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arXiv2020

Single and multiple recurrence along non-polynomial sequences

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $\ell\in \mathbb{N}\cup\{0\}$, $\lim_{x\to\infty }f^{(\ell)}(x)=\pm\infty$ and $\lim_{x\to\infty }f^{(\ell+1)}(x)=0$. Among other things, we show that for any $f\in\mathcal{F}$, any invertible probability measure preserving system $(X,\mathcal{B},μ,T)$, any $A\in\mathcal{B}$ with $μ(A)>0$, and any $ε>0$, the sets of returns $$ R_{ε, A}= \big\{n\in\mathbb{N}:μ(A\cap T^{-\lfloor f(n)\rfloor}A)>μ^2(A)-ε\big\} $$ and $$ R^{(k)}_{A}= \big\{ n\in\mathbb{N}: μ\big(A\cap T^{\lfloor f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor f(n+k)\rfloor}A\big)>0\big\} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.

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arXiv2020

A fresh look at the notion of normality

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) Følner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the $(F_n)$-normality coincides with the classical notion. We prove that: $\bullet$ If $(F_n)$ is a Følner sequence in $G$, such that for every $α\in(0,1)$ we have $\sum_n α^{|F_n|}<\infty$, then almost every $x\in\{0,1\}^G$ is $(F_n)$-normal. $\bullet$ For any Følner sequence $(F_n)$ in $G$, there exists an Cham\-per\-nowne-like $(F_n)$-normal set. $\bullet$ There is a natural class of "nice" Følner sequences in $(\mathbb N,\times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice Følner \sq. $\bullet$ Let $A\subset\mathbb N$ be a classical normal set. Then, for any Følner sequence $(K_n)$ in $(\mathbb N,\times)$ there exists a set $E$ of $(K_n)$-density $1$, such that for any finite subset $\{n_1,n_2,\dots,n_k\}\subset E$, the intersection $A/{n_1}\cap A/{n_2}\cap\ldots\cap A/{n_k}$ has positive upper density in $(\mathbb N,+)$. As a consequence, $A$ contains arbitrarily long geometric progressions, and, more generally, arbitrarily long "geo-arithmetic" configurations of the form $\{a(b+ic)^j,0\le i,j\le k\}$. $\bullet$ For any Følner \sq\ $(F_n)$ in $(\mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $\bullet$ For any nice Følner sequence $(F_n)$ in $(\mathbb N,\times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.

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arXiv2020

Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

A classical Kamae-Weiss theorem states that an increasing sequence $(n_i)_{i\in\mathbb N}$ of positive lower density is \emph{normality preserving}, i.e. has the property that for any normal binary sequence $(b_n)_{n\in\mathbb N}$, the sequence $(b_{n_i})_{i\in\mathbb N}$ is normal, if and only if $(n_i)_{i\in\mathbb N}$ is a deterministic sequence. Given a countable cancellative amenable semigroup $G$, and a Følner sequence $\mathcal F=(F_n)_{n\in\mathbb N}$ in $G$, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of $G$ with respect to $\mathcal F$, and show that for sets of positive lower $\mathcal F$-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on $\mathcal F$, positive lower $\mathcal F$-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups

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arXiv2019

An extension of Weyl's equidistribution theorem to generalized polynomials and applications

Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) λ)_{n \in \mathbb{Z}}$ is well distributed $\bmod \, 1$ for all but countably many $λ\in \mathbb{R}$ if and only if $\lim\limits_{\substack{|n| \rightarrow \infty n \notin J}} |q(n)| = \infty$ for some (possibly empty) set $J$ having zero density in $\mathbb{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of I. Vinogradov and G. Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

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arXiv2019

A structure theorem for level sets of multiplicative functions and applications

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let $E=\{n_1 0$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$, $i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N μ\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$ We also show that if a level set $E$ of a multiplicative function has positive upper density, then any self-shift $E-r$, $r\in E$, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemerédi theorem (cf. [4]). Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$ has positive upper density and let $r\in E$. Then for any set $D\subset \mathbb{N}$ with positive upper density and any polynomials $p_i\in\mathbb{Q}[t]$, $i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>β\,\right\} $$ has positive lower density.

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