A structure theorem for sets of small popular doubling

Przemysław Mazur

Displaying similar documents to “A structure theorem for sets of small popular doubling”

Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒫_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$ , $q$ such that $(a, q) = 1$ , let $𝒫_{2} (q, a)$ denote the least $𝒫_{2}$ in the arithmetic progression ${n q + a}_{n = 1}^{\infty}$ . It is proved that for sufficiently large $q$ , we have $𝒫_{2} (q, a) ≪ q^{1.825} .$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $𝒫_{2} (q, a) ≪ q^{1.8345} .$

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

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A geometric progression of length k and integer ratio is a set of numbers of the form $a, a r, . . ., a r^{k - 1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${(a_{i})}_{i = 1}^{\infty}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃_{i = 1}^{\infty} (a_{2 i}, a_{2 i - 1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper, we give a numerical characterization of nef arithmetic $ℝ$ -Cartier divisors of $C^{0}$ -type on an arithmetic surface. Namely an arithmetic $ℝ$ -Cartier divisor $\bar{D}$ of $C^{0}$ -type is nef if and only if $\bar{D}$ is pseudo-effective and $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

Coprimality of integers in Piatetski-Shapiro sequences

Watcharapon Pimsert, Teerapat Srichan, Pinthira Tangsupphathawat (2023)

Czechoslovak Mathematical Journal

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We use the estimation of the number of integers $n$ such that $⌊ n^{c} ⌋$ belongs to an arithmetic progression to study the coprimality of integers in $ℕ^{c} = {⌊ n^{c} ⌋}_{n \in ℕ}$ , $c > 1$ , $c \notin ℕ$ .

The Golomb space is topologically rigid

Taras O. Banakh, Dario Spirito, Sławomir Turek (2021)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space $ℕ_{τ}$ is the set $ℕ$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions ${a + b n : n \geq 0}$ with coprime $a, b$ . We prove that the Golomb space $ℕ_{τ}$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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We consider random walk on a discrete torus $E$ of side-length $N$ , in sufficiently high dimension $d$ . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u N^{d}$ . We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $log N$ . Moreover, this connected component occupies a...

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Positivity of integrated random walks

Vladislav Vysotsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Suppose $S_{1}$ is in the domain of normal attraction of an $α$ -stable law with $1 l t; α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} g t; x | S_{1} g t; 0) = e^{- a x}$ for some $a g t; 0$ and all $x g t; 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} g t; 0, \dots, A_{N} g t; 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} g t; 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

On a sum involving the integral part function

Bo Chen (2024)

Czechoslovak Mathematical Journal

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Let $[t]$ be the integral part of a real number $t$ , and let $f$ be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum $S_{f} (x) = \sum_{n \leq x} f ([x / n])$ , which improves the recent result of J. Stucky (2022).

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper, we study the size of the giant component $C_{G}$ in the random geometric graph $G = G (n, r_{n}, f)$ of $n$ nodes independently distributed each according to a certain density $f (\cdot)$ in ${[0, 1]}^{2}$ satisfying ${inf}_{x \in {[0, 1]}^{2}} f (x) g t;$ $0$ . If $\frac{c_{1}}{n} \leq r_{n}^{2} \leq c_{2} \frac{log n}{n}$ for some positive constants $c_{1}$ , $c_{2}$ and $n r_{n}^{2} ⟶ \infty$ as $n \to \infty$ , we show that the giant component of $G$ contains at least $n - o (n)$ nodes with probability at least $1 - e^{- β n r_{n}^{2}}$ for all $n$ and for some positive constant $β$ . We also obtain estimates on the diameter and number of the non-giant components of $G$ .

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

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We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.