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

K. P. S. Bhaskara Rao; Vincenzo Aversa

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

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

Euline Green (1971)

Colloquium Mathematicae

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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.

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.

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.

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 $f (X)$ .

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.

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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Semifields and a theorem of Abhyankar

Vítězslav Kala (2017)

Commentationes Mathematicae Universitatis Carolinae

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Abhyankar proved that every field of finite transcendence degree over $ℚ$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $ℤ [T_{1}, \dots, T_{n}]$ (for some $n$ depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

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 ${| p \cdot A + q \cdot A | \geq (p + q) | A | - (p q)}^{(p + q - 3) (p + q) + 1}$ .

$F_{σ}$ -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.

$L_{1}$ contains every two-dimensional normed space

David Yost (1988)

Annales Polonici Mathematici

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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...

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

Alex Gamburd, Dmitry Jakobson, Peter Sarnak (1999)

Journal of the European Mathematical Society

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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...

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.

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

On the concentration of certain additive functions

Dimitris Koukoulopoulos (2014)

Acta Arithmetica

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We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of $f (n) = \sum_{p | n} {(l o g p)}^{- c}$ when c > 1.

Effective results for Diophantine equations over finitely generated domains

Attila Bérczes, Jan-Hendrik Evertse, Kálmán Győry (2014)

Acta Arithmetica

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Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f (x) = δ y^{m}$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and $m \in ℤ_{\geq 2}$ . Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F,...

Displaying similar documents to “A remark on E. Green’s paper “Completeness of L p -spaces over finitely additive set functions””

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