On the concentration of certain additive functions

Dimitris Koukoulopoulos

Displaying similar documents to “On the concentration of certain additive functions”

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.

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.

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.

On a conjecture of Sárközy and Szemerédi

Yong-Gao Chen (2015)

Acta Arithmetica

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Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(minA(x),B(x)), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given...

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 $f (X)$ .

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 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.

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

Euline Green (1971)

Colloquium Mathematicae

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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

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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. ...

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.

$F_{σ}$ -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 Vitali-Hahn-Saks-Nikodym type theorems

Barbara T. Faires (1976)

Annales de l'institut Fourier

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A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $(a_{n})$ , $(b_{m})$ in $𝒜$ with $a_{n} \leq b_{m}$ for all $n, m$ , there exists an element $b$ in $𝒜$ such that $a_{n} \leq b \leq b_{n}$ for all $n$ . Let $𝒜$ denote an algebra with the property (I). It is shown that if $(μ_{n} : 𝒜 \to X)$ ( $X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_{n} μ_{n} (a)$ exists for each $a \in 𝒜$ , then $μ (a) = {lim}_{n} μ_{n} (a)$ defines a strongly additive map from $𝒜$ to $X$ the $μ_{n}^{'} s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive...

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

Ben Green (2001)

Acta Arithmetica

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New results on additive generator pairs of overlap and grouping functions

Liang Li-zhi, Wang Xue-ping (2025)

Kybernetika

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In this article, we deeply reveal the relationship between functions $θ$ and $ϑ$ in an overlap function additively generated by an additive generator pair ( $θ$ , $ϑ$ ), which is used to characterize the conditions for an overlap function additively generated by the pair being a triangular norm by terms of functions $θ$ and $ϑ$ . We also establish the conditions that an overlap function additively generated by the additive generator pair can be obtained by a distortion of a triangular norm and a (pseudo)...

Unique factorisation of additive induced-hereditary properties

Alastair Farrugia, R. Bruce Richter (2004)

Discussiones Mathematicae Graph Theory

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An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ₁,...,ₙ be additive hereditary graph properties. A graph G has property (₁∘...∘ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph $G [V_{i}]$ is in $_{i}$ . A property is reducible if there are properties , such that = ∘ ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive...