On Meager Additive and Null Additive Sets in the Cantor Space $2^{ω}$ and in ℝ

Tomasz Weiss

Displaying similar documents to “On Meager Additive and Null Additive Sets in the Cantor Space $2^{ω}$ and in ℝ”

Addendum to “On Meager Additive and Null Additive Sets in the Cantor space $2^{ω}$ and in ℝ” (Bull. Polish Acad. Sci. Math. 57 (2009), 91-99)

Tomasz Weiss (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We prove in ZFC that there is a set $A \subseteq 2^{ω}$ and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), $H^{- 1} (X)$ is null additive in $2^{ω}$ . This settles in the affirmative a question of T. Bartoszyński.

A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Fateme Kouchakinejad, Alexandra Šipošová (2017)

Kybernetika

Similarity:

For an aggregation function $A$ we know that it is bounded by $A^{*}$ and $A_{*}$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^{*}$ is directionally convex, then $A = A^{*}$ and $A_{*}$ is linear; similarly, if $A_{*}$ is directionally concave, then $A = A_{*}$ and $A^{*}$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively. ...

Completeness of $L^{p}$ -spaces over finitely additive set functions

Euline Green (1971)

Colloquium Mathematicae

Similarity:

Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

Similarity:

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.

On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Consider the power series $(z) = \sum_{n = 1}^{\infty} α (n) z ⁿ$ , where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity $e^{2 π i l / q}$ . We give effective omega-estimates for $(e (l / p^{k}) r)$ when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

The number of squares and $B_{h} [g]$ sets

Ben Green (2001)

Acta Arithmetica

Similarity:

On the sum of dilations of a set

Antal Balog, George Shakan (2014)

Acta Arithmetica

Similarity:

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

On strong measure zero subsets of $^{κ} 2$

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

Similarity:

We study the generalized Cantor space $^{κ} 2$ and the generalized Baire space $^{κ} κ$ as analogues of the classical Cantor and Baire spaces. We equip $^{κ} κ$ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of $^{κ} 2$ . We prove for successor $κ = κ^{< κ}$ that the ideal of strong measure zero sets of $^{κ} 2$ is $_{κ}$ -additive, where $_{κ}$ is the size of the smallest unbounded family in $^{κ} κ$ , and that the generalized Borel...

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

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

Similarity:

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.

$F_{σ}$ -additive covers of Čech complete and scattered-K-analytic spaces

Jiří Spurný (2008)

Fundamenta Mathematicae

Similarity:

We prove that an $F_{σ}$ -additive cover of a Čech complete, or more generally scattered-K-analytic space, has a σ-scattered refinement. This generalizes results of G. Koumoullis and R. W. Hansell.

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{α}$ , $Y_{α}$ and $Z_{α}$ such that (i) $f X_{α}, f Y_{α}, f Z_{α} = ω ₀$ , where f is either trdef or ₀-trsur, (ii) $A (α) - t r i n d X_{α} = \infty$ and $M (α) - t r i n d X_{α} = - 1$ , (iii) $A (α) - t r i n d Y_{α} = - 1$ and $M (α) - t r i n d Y_{α} = \infty$ , and (iv) $A (α) - t r i n d Z_{α} = M (α) - t r i n d Z_{α} = \infty$ and $A (α + 1) \cap M (α + 1) - t r i n d Z_{α} = - 1$ . We also show that there exists no separable metrizable space $W_{α}$ with $A (α) - t r i n d W_{α} \neq \infty$ , $M (α) - t r i n d W_{α} \neq \infty$ and $A (α) \cap M (α) - t r i n d W_{α} = \infty$ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

Some Additive $2 - (v, 5, λ)$ Designs

Andrea Caggegi (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Similarity:

Given a finite additive abelian group $G$ and an integer $k$ , with $3 \leq k \leq | G |$ , denote by $𝒟_{k} (G)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$ -subsets $C = {c_{1}, c_{2}, \dots, c_{k}}$ of $G$ such that $c_{1} + c_{2} + \dots + c_{k} = 0$ . It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that $𝒟_{k} (G)$ is a 2-design, if $G$ is an elementary abelian $p$ -group with $p$ a prime divisor of $k$ . From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity...

Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_{r}$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n \geq 1$ a fixed positive integer. Let $α$ be an automorphism of the ring $R$ . An additive map $D : R \to R$ is called an $α$ -derivation (or a skew derivation) on $R$ if $D (x y) = D (x) y + α (x) D (y)$ for all $x, y \in R$ . An additive mapping $F : R \to R$ is called a generalized $α$ -derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F (x y) = F (x) y + α (x) D (y)$ for all $x, y \in R$ . We prove...

A remark on E. Green’s paper “Completeness of $L^{p}$ -spaces over finitely additive set functions”

K. P. S. Bhaskara Rao, Vincenzo Aversa (1987)

Colloquium Mathematicae

Similarity:

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

Similarity:

Given a set A ⊂ ℕ let $σ_{A} (n)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_{A} (n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.

Goldbach’s problem with primes in arithmetic progressions and in short intervals

Karin Halupczok (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n \neg \equiv 1 (6)$ , Goldbach’s ternary problem $n = p_{1} + p_{2} + p_{3}$ is solvable with primes $p_{1}, p_{2}$ in short intervals $p_{i} \in [X_{i}, X_{i} + Y]$ with $X_{i}^{θ_{i}} = Y$ , $i = 1, 2$ , and $θ_{1}, θ_{2} \geq 0.933$ such that $(p_{1} + 2) (p_{2} + 2)$ has at most $9$ prime factors.

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

Similarity:

Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

Similarity:

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

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

Similarity:

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K (t), i (t), Y (t))$ on $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , where $𝕋^{2}$ is the two-dimensional torus. Here $(K (t), i (t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y (t)$ is an additive functional of $K$ , defined as $\int_{0}^{t} v (K (s)) d s$ , where $| v | \sim 1$ for small $k$ . We prove that the rescaled process ${(N ln N)}^{- 1 / 2} Y (N t)$ converges in distribution to a two-dimensional Brownian motion. As a consequence,...

Eigenfunctions of the Laplace Operators for a Building of Type $\widetilde{G}_{2}$

A. M. Mantero, A. Zappa (2009)

Bollettino dell'Unione Matematica Italiana

Similarity:

Let $\Delta$ be thick affine building of type $\widetilde{G}_{2}$ the Laplace operators of $\Delta$ ; associated with a pair $(y_{1},y_{2})$ ; is the Poisson transform of a suitable finitely additive measure on the maximal boundary $\Omega$ of $\Delta$ ; by using only the combinatorial structure of $\Delta$ .

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

Similarity:

We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Spectra of elements in the group ring of SU(2)

Alex Gamburd, Dmitry Jakobson, Peter Sarnak (1999)

Journal of the European Mathematical Society

Similarity:

We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU (2)$ , providing an elementary solution of Ruziewicz problem on $S^{2}$ as well as giving many new examples of finitely generated subgroups of $SU (2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $𝐑 [SU (2)]$ in the $N$ -th irreducible representation of $SU (2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $𝐑 [SU (2)]$ , the “unfolded” consecutive...

Displaying similar documents to “On Meager Additive and Null Additive Sets in the Cantor Space 2 ω and in ℝ”

Displaying similar documents to “On Meager Additive and Null Additive Sets in the Cantor Space $2^{ω}$ and in ℝ”