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Some sufficient conditions for zero asymptotic density and the expression of natural numbers as sum of values of special functions

Pavel Jahoda, Monika Pěluchová (2005)

Acta Mathematica Universitatis Ostraviensis

This paper generalizes some results from another one, namely [3]. We have studied the issues of expressing natural numbers as a sum of powers of natural numbers in paper [3]. It means we have studied sets of type A = { n 1 k 1 + n 2 k 2 + ⋯ + n m k m ∣ n i ∈ ℕ ∪ { 0 } , i = 1 , 2 ⋯ , m , ( n 1 , n 2 , ⋯ , n m ) ≠ ( 0 , 0 , ⋯ , 0 ) } , where k 1 , k 2 , ⋯ , k m ∈ ℕ were given natural numbers. Now we are going to study a more general case, i.e. sets of natural numbers that are expressed as sum of integral parts of functional values of some special functions. It means that we are interested in sets of natural numbers in the form k = [ f 1 ( n 1 ) ] + [ f 2 ( n 2 ) ] + ⋯ + [ f m ( n m ) ] .

Sum-dominant sets and restricted-sum-dominant sets in finite abelian groups

David B. Penman, Matthew D. Wells (2014)

Acta Arithmetica

We call a subset A of an abelian group G sum-dominant when |A+A| > |A-A|. If |A⨣A| > |A-A|, where A⨣A comprises the sums of distinct elements of A, we say A is restricted-sum-dominant. In this paper we classify the finite abelian groups according to whether or not they contain sum-dominant sets (respectively restricted-sum-dominant sets). We also consider how much larger the sumset can be than the difference set in this context. Finally, generalising work of Zhao, we provide asymptotic estimates...

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

We describe all sets A ⊆ p which represent the quadratic residues R ⊆ p in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

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