On Exponential Densities and Limit Ratios of Subsets of $${\mathbb {N}}$$ N
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On Exponential Densities and Limit Ratios of Subsets of N J. Li and L. Olsen Abstract. Given α, β, γ ∈ [0, 1] with α ≤ β, we prove that there exists a subset of N such that its lower and upper exponential densities and its lower and upper limit ratios are equal to α, β, γ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011). Mathematics Subject Classification. Primary 11B05. Keywords. Positive integer sequence, Exponential densities, Limit ratios.
1. Introduction and Statement of Results The purpose of this paper is to provide an affirmative solution to a problem posed by Grekos et al. [5] about the interrelationship between the exponential densities and the limit ratios of subsets of integers. We start by recalling the definitions of exponential densities and limit ratios of subsets of integers. Definition. (Exponential densities) For an infinite subset A of N, write Nn (A) = {1, . . . , n} ∩ A ; here and below, we write |A| for the cardinality of a set A. We define the lower and upper exponential densities of an infinite subset A of N by ε(A) = lim inf
log Nn (A) log n
ε(A) = lim sup
log Nn (A) , log n
n
and n
respectively. Definition. (Limit ratios) For an infinite subset A = {a1 , a2 , . . .} of N with a1 < a2 < · · · , we define the lower and upper limit ratios of A by an (A) = lim inf n an+1 0123456789().: V,-vol
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and (A) = lim sup n
an , an+1
respectively. The exponential densities and the limit ratios are fundamental in many diverse areas of pure and applied mathematics, including analytic number theory [1,11], metric number theory [4,6,7], and more recently in fractal geometry of discrete sets [2,9] and theoretical computer science [3]; the reader is referred to the remarks below for more details. Because of their important and ubiquitous role, it is natural to investigate their interrelationship. For example, Grekos et al. [5] proved that if (A) < 1, then ε(A) = 0, and they also present examples showing that if (A) = 1, then nothing can be said about the value of ε(A). This observation led Grekos et al. [5] to ask the following question. Question. [5, Problem 2.7] Given α, β, γ ∈ [0, 1] with α ≤ β, does there exist an infinite subset A of N such that ε(A) = α , ε(A) = β , (A) = γ , (A) = 1 ? The main purpose of this paper is to provide an affirmative answer to this question. We formally state our result as follows. Theorem 1.1. Given α, β, γ ∈ [0, 1] with α ≤ β, then there exists an infinite subset A of N such that ε(A) = α, ε(A) = β, (A) = γ, (A) = 1 . The proof of Theorem 1.1 is given in Sects. 2–4; Sect. 2 contains some auxiliary results; in Sect. 3 we prove Theorem 1.1 for β = 0; and in Sect. 4 we prove Theorem 1.1 for β > 0. The main difficulty in the proof of Theorem 1.1 is to ensure that (A) = γ. Indeed, it is not difficult to see that there is a subset A of N with ε(A) = α and ε(A) = β. To see this we note that it is not difficult to show that we can choose po
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