Erratum to “Magic $p$-dimensional cubes” (Acta Arith. 96 (2001), 361–364)

2019 ◽  
Vol 191 (1) ◽  
pp. 95-100
Author(s):  
C. H. Silva de Souza ◽  
J. N. Tomazella
Keyword(s):  
Acta Arith

SOME REMARKS ON MINIMAL ASYMPTOTIC BASES OF ORDER THREE

2020 ◽  
Vol 102 (1) ◽  
pp. 21-30
Author(s):  
DENGRONG LING ◽  
MIN TANG
Keyword(s):  
Acta Arith ◽  
Powers Of 2

We study a question on minimal asymptotic bases asked by Nathanson [‘Minimal bases and powers of 2’, Acta Arith. 49 (1988), 525–532].

scholarly journals ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

2011 ◽  
Vol 53 (3) ◽  
pp. 669-681
Author(s):  
NATALIA BUDARINA
Keyword(s):  
Acta Arith ◽  
Algebraic Numbers ◽  
Natural Restriction ◽  
Probabilistic Criterion ◽  
Error Functions ◽  
System Of Inequalities ◽  
Compositio Math ◽  
General Error

AbstractIn this paper, the Khintchine-type theorems of Beresnevich (Acta Arith.90(1999), 97) and Bernik (Acta Arith.53(1989), 17) for polynomials are generalised to incorporate a natural restriction on derivatives. This represents the first attempt to solve a problem posed by Bernik, Kleinbock and Margulis (Int. Math. Res. Notices2001(9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities |P(x)| < Ψ1(H) and |P′(x)| < Ψ2(H) in integral polynomialsPof degree ≤nand heightH, where Ψ1and Ψ2are fairly general error functions. The proof builds upon Sprindzuk's method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Götze (Compositio Math.146(2010), 1165) concerning the distribution of algebraic numbers.

Quasi-uniqueness of the set of "Gaussian prime plus one's"

2014 ◽  
Vol 10 (07) ◽  
pp. 1783-1790
Author(s):  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Acta Arith ◽  
Gaussian Integers ◽  
Arithmetical Functions ◽  
Set Of Uniqueness ◽  
Positive Integers ◽  
Complex Valued ◽  
Gaussian Primes

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

scholarly journals Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size

2020 ◽  
Vol 15 (1) ◽  
pp. 1-26
Author(s):  
Pierre-Adrien Tahay
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Correlation Coefficients ◽  
Large Family ◽  
Acta Arith ◽  
Pseudorandom Sequences ◽  
Arbitrary Size ◽  
Difference Matrices ◽  
Prime Factors

AbstractIn 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

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