# Tom 15 (2020): Zeszyt 2 (December 2020)

Informacje o czasopiśmie
Format
Czasopismo
eISSN
2309-5377
Pierwsze wydanie
30 Dec 2013
Częstotliwość wydawania
2 razy w roku
Języki
Angielski

7 Artykułów
Otwarty dostęp

#### A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

Data publikacji: 25 Dec 2020
Zakres stron: 1 - 8

#### Abstrakt

Let α be an irrational real number; the behaviour of the sum SN (α):= (1)[α] +(1)[2α] + ··· +(1)[] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of 2/2\sqrt 2 /2 has bounded partial quotients, SN(2)=O(log(N)){S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus SN(2e)=O(log(N)2loglog(N)2){S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough SN(e)=O(log(N)loglog(N))1188.

#### Słowa kluczowe

• Oscillating sums
• uniform distribution modulo 1

#### MSC 2010

• Primary 11K38
• Secondary 11A55
Otwarty dostęp

#### On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

Data publikacji: 25 Dec 2020
Zakres stron: 9 - 22

#### Abstrakt

Both the Thue–Morse and Rudin–Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue–Morse sequence along squares keeps a large maximum order complexity. Since, by Christol’s theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin–Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems.

#### Słowa kluczowe

• Automatic sequences
• pseudorandomness
• Thue–Morse sequence
• Rudin–Shapiro sequence
• polynomials

• 11A63
• 11B85
Otwarty dostęp

#### Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Data publikacji: 25 Dec 2020
Zakres stron: 23 - 38

#### Abstrakt

Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, xX and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.

#### Słowa kluczowe

• Stationary measure
• Minkowski’s question matk function
• Word metric
• Lattice orbits

• 22E40
• 30B70
• 60G10
Otwarty dostęp

#### On Proinov’s Lower Bound for the Diaphony

Data publikacji: 25 Dec 2020
Zakres stron: 39 - 72

#### Abstrakt

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

#### Słowa kluczowe

• ℒ-discrepancy
• Walsh system
• Haar system

• 11K38
Otwarty dostęp

#### The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

Data publikacji: 25 Dec 2020
Zakres stron: 73 - 92

#### Abstrakt

We give a heuristic argument predicting that the number N(T) of rationals p/q on Cantor’s middle thirds set C such that gcd(p, q)=1 and q ≤ T, has asymptotic growth O(Td+ε), for d = dim C. Our heuristic is related to similar heuristics and conjectures proposed by Fishman and Simmons. We also describe extensive numerical computations supporting this heuristic. Our heuristic predicts a similar asymptotic if C is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of N (T)is motivated by a problem of Mahler on intrinsic Diophantine approximation on C.

#### Słowa kluczowe

• Rational numbers in the Cantor set

#### MSC 2010

• 11K60: Diophantine approximation in probabilistic number theory
Otwarty dostęp

#### Point Distribution and Perfect Directions in $𝔽p2$\mathbb{F}_p^2

Data publikacji: 25 Dec 2020
Zakres stron: 93 - 98

#### Abstrakt

Let p ≥ 3 be a prime, S𝔽p2S \subseteq \mathbb{F}_p^2 a nonempty set, and w:𝔽p2Rw:\mathbb{F}_p^2 \to R a function with supp w = S. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most 12|S|{1 \over 2}\left| S \right| directions in 𝔽p2\mathbb{F}_p^2 such that for every line l in any of these directions, one has zlw(z)=1pz𝔽p2w(z),\sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} } except if S itself is a line and w is constant on S (in which case all, but one direction have the property in question). The bound 12|S|{1 \over 2}\left| S \right| is sharp.

As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.

#### Słowa kluczowe

• Uniform distribution
• affine plane

#### MSC 2010

• Primary: 05B25
• Secondary: 51E99
Otwarty dostęp

#### On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

Data publikacji: 25 Dec 2020
Zakres stron: 99 - 112

#### Abstrakt

Motivated by the maximal average distance of uniformly distributed sequences we consider some extremal problems for functionals of type μC0101FdμC,{\mu _C} \mapsto \int_0^1 {{{\int_0^1 {Fd} }_\mu }_C,} where µC is a copula measure and F is a Riemann integrable function on [0, 1]2 of a specific type. Such problems have been considered in [4] and are of interest in the study of limit points of two uniformly distributed sequences.

#### Słowa kluczowe

• Uniform distribution
• Copulas
• Extremal problems

• 11K06
• 62H05