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

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Displaying similar documents to “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)”

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

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let T be the standard Cantor-Lebesgue function that maps the Cantor space $2^{ω}$ onto the unit interval ⟨0,1⟩. We prove within ZFC that for every $X \subseteq 2^{ω}$ , X is meager additive in $2^{ω}$ iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in $2^{ω}$ and ℝ.

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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. ...

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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.

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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.

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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...

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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.

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Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.

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

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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...

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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...

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

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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,...

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

Displaying similar documents to “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)”

Displaying similar documents to “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)”