Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck; I. Kátai

Displaying similar documents to “Sums of reciprocals of additive functions running over short intervals”

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.

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.

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

Similarity:

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.

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:

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

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

Tomasz Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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.

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

Ondřej Zindulka (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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 $f (X)$ .

Polynomial quotients: Interpolation, value sets and Waring's problem

Zhixiong Chen, Arne Winterhof (2015)

Acta Arithmetica

Similarity:

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p, w} (u)$ are defined by $q_{p, w} (u) \equiv (u^{w} - u^{w p}) / p m o d p$ with $0 \leq q_{p, w} (u) \leq p - 1$ , u ≥ 0, which are generalizations of Fermat quotients $q_{p, p - 1} (u)$ . First, we estimate the number of elements $1 \leq u < N \leq p$ for which $f (u) \equiv q_{p, w} (u) m o d p$ for a given polynomial f(x) over the finite field $_{p}$ . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...

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.

On an additive problem of unlike powers in short intervals

Qingqing Zhang (2022)

Czechoslovak Mathematical Journal

Similarity:

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 $1 \leq k \leq 5$ .

Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi (2021)

Czechoslovak Mathematical Journal

Similarity:

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 $3$ .

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

Ben Green (2001)

Acta Arithmetica

Similarity:

Ultrafilter extensions of asymptotic density

Jan Grebík (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

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.

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...

On sum-product representations in $ℤ_{q}$

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

Similarity:

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$ , by performing sum and product set operations starting from a given subset $A$ of $ℤ_{q}$ . We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $| A | \sim log q$ ). Roughly speaking we show that all residue classes are obtained from a $k$ -fold sum of an $r$ -fold product set of $A$ , where $r ≪ log q$ and $log k ≪ log q$ , provided the...

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

Similarity:

Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q (m)$ , where $Q$ is a polynomial with integral coefficients, then $P (x) = Q (R (x))$ for some polynomial $R$ . In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^{q}$ where $q$ is an integer greater than 1, then $P (x) = {(R (x))}^{q}$ for some polynomial $R (x)$ .

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

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

Communications in Mathematics

Similarity:

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.