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

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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

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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. ...

Strong measure zero and meager-additive sets through the prism of fractal measures

Ondřej Zindulka (2019)

Commentationes Mathematicae Universitatis Carolinae

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We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{ω}$ is meager-additive if and only if it is $ℰ$ -additive; if $f : 2^{ω} \to 2^{ω}$ is continuous and $X$ is meager-additive, then so is $f (X)$ .

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

Euline Green (1971)

Colloquium Mathematicae

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More remarks on the intersection ideal $ℳ \cap 𝒩$

Tomasz Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove in ZFC that every $ℳ \cap 𝒩$ additive set is $𝒩$ additive, thus we solve Problem 20 from paper [Weiss T., A note on the intersection ideal $ℳ \cap 𝒩$ , Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445] in the negative.

Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

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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.

On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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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.

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

Ben Green (2001)

Acta Arithmetica

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Ultrafilter extensions of asymptotic density

Jan Grebík (2019)

Commentationes Mathematicae Universitatis Carolinae

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We characterize for which ultrafilters on $ω$ is the ultrafilter extension of the asymptotic density on natural numbers $σ$ -additive on the quotient boolean algebra $𝒫 (ω) / d_{𝒰}$ or satisfies similar additive condition on $𝒫 (ω) / fin$ . These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name $A P$ (null) and $A P$ (*). We also present a characterization of a $P$ - and semiselective...

On the sum of dilations of a set

Antal Balog, George Shakan (2014)

Acta Arithmetica

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

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

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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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...

Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi (2021)

Czechoslovak Mathematical Journal

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Let $𝔽_{q} [t]$ denote the polynomial ring over $𝔽_{q}$ , the finite field of $q$ elements. Suppose the characteristic of $𝔽_{q}$ is not $2$ or $3$ . We prove that there exist infinitely many $N \in ℕ$ such that the set ${f \in 𝔽_{q} [t] : deg f < N}$ contains a Sidon set which is an additive basis of order $3$ .

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

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

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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.

Nonlinear $*$ -Lie higher derivations of standard operator algebras

Mohammad Ashraf, Shakir Ali, Bilal Ahmad Wani (2018)

Communications in Mathematics

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Let $ℋ$ be an infinite-dimensional complex Hilbert space and $𝔄$ be a standard operator algebra on $ℋ$ which is closed under the adjoint operation. It is shown that every nonlinear $*$ -Lie higher derivation $𝒟 = {δ_{n}}_{n \in ℕ}$ of $𝔄$ is automatically an additive higher derivation on $𝔄$ . Moreover, $𝒟 = {δ_{n}}_{n \in ℕ}$ is an inner $*$ -higher derivation.

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

Jiří Spurný (2008)

Fundamenta Mathematicae

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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.

On an additive problem of unlike powers in short intervals

Qingqing Zhang (2022)

Czechoslovak Mathematical Journal

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We prove that almost all positive even integers $n$ can be represented as $p_{2}^{2} + p_{3}^{3} + p_{4}^{4} + p_{5}^{5}$ with $| p_{k}^{k} - \frac{1}{4} N | \leq N^{1 - 1 / 54 + ε}$ for $2 \leq k \leq 5$ . As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_{1} + p_{2}^{2} + p_{3}^{3} + p_{4}^{4} + p_{5}^{5}$ with $| p_{k}^{k} - \frac{1}{5} N | \leq N^{1 - 1 / 54 + ε}$ for $1 \leq k \leq 5$ .

Infinite-Dimensionality modulo Absolute Borel Classes

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.

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”