Let h ≥ 2 be an integer. We say that a set A of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from A. A set of positive integers A is called a Sidon set if all the sums a + b with a, b ∈ A, a ≤ b are distinct. In this paper we prove the existence of Sidon set A which is an asymptotic basis of order 4 by using probabilistic methods.
2000See [7] for the proof. Informally this theorem asserts that when the derivatives of Y are smaller on average than Y itself, and the degree of Y is small, then Y is concentrated around its mean. Finally we need the Borel -Cantelli lemma:Lemma 3. (Borel-Cantelli) Let X 1 , X 2 , . . . be a sequence of events in a probability space. If +∞ j=1 P(X j ) < ∞, then with probability 1, at most a finite number of the events X j can occur.See in [6], p. 135.
Let X be a semigroup written additively and h ≥ 2 a fixed integer. Let x be an element of X and A1, . . . , A h be nonempty subsets of X.denote the number of solutions of the equation a1 + • • • + a h = x, where ai ∈ Ai. In this paper for X = N we give a necessary and sufficient condition such that the equalityholds from a certain point on. We study similar questions when X = Zm and in general when X = G, where G is a finite additive group.
In this paper we investigate how small the density of a multiplicative basis of order h can be in {1, 2, . . . , n} and in Z + . Furthermore, a related problem of Erdős is also studied: How dense can a set of integers be, if none of them divides the product of h others?
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