Set of uniqueness of shifted 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].

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Set of Uniqueness on Noncommutative Locally Compact Groups

Proceedings of the American Mathematical Society ◽

10.2307/2040989 ◽

1977 ◽

Vol 64 (1) ◽

pp. 93 ◽

Cited By ~ 2

Author(s):

Marek Bozejko

Keyword(s):

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Set Of Uniqueness

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Chapter 12. Variations on Gaussian Primes

Prime-Detecting Sieves. (LMS-33) ◽

10.1515/9781400845934.265 ◽

2012 ◽

pp. 265-302

Keyword(s):

Gaussian Primes

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The Gaussian primes

The Theory of Algebraic Numbers ◽

10.5948/9781614440093.004 ◽

1975 ◽

pp. 14-24

Keyword(s):

Gaussian Primes

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Perfect Sets of Uniqueness on the Group 2ω

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-052-1 ◽

1982 ◽

Vol 34 (3) ◽

pp. 759-764 ◽

Cited By ~ 7

Author(s):

Kaoru Yoneda

Keyword(s):

Haar Measure ◽

Measure Zero ◽

Countable Subset ◽

Walsh Series ◽

Dyadic Group ◽

Set Of Uniqueness ◽

Image Position ◽

Sets Of Uniqueness

Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfiesA subset E of G which is not a set of uniqueness is called a set of multiplicity.It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

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