8-1-2014 Four Quotient Set Gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years. INTRODUCTION. If A is a subset of the natural numbers N = {1, 2, . . .}, then we let R(A) = {a/a : a, a ∈ A} denote the corresponding quotient set (sometimes called a ratio set). Our aim in this short note is to present four remarkable results, which seem to have been overlooked in the MONTHLY, despite its intense coverage of quotient sets over the years [3, 4, 5, 9, 10, 11, 14]. Some of these results are novel, while others have appeared in print elsewhere but somehow remain largely unknown. In what follows, we let A(x) = A ∩ [1, x] so that |A(x)| denotes the number of elements in A which are ≤ x . The lower asymptotic density of A is the quantity d(A) = lim inf n→∞ |A(n)| n , which satisfies the obvious bounds 0 ≤ d(A) ≤ 1. We say that A is fractionally dense if the closure of R(A) in R equals [0,∞) (i.e., if R(A) is dense in [0,∞)). Our four gems are as follows. 1. The set of all natural numbers whose base-b representation begins with the digit 1 is fractionally dense for b = 2, 3, 4, but not for b ≥ 5. 2. For each δ ∈ [0, 12 ), there exists a set A ⊂ N with d(A) = δ that is not fractionally dense. On the other hand, if d(A) ≥ 12 , then A must be fractionally dense [15]. 3. We can partition N into three sets, each of which is not fractionally dense. However, such a partition is impossible using only two sets [2]. 4. There are subsets of N that contain arbitrarily long arithmetic progressions, yet that are not fractionally dense. On the other hand, there exist fractionally dense sets that have no arithmetic progressions of length ≥ 3. BASE-b REPRESENTATIONS. In [5, Example 19], it was shown that the set A = {1} ∪ {10, 11, 12, 13, 14, 15, 16, 17, 18, 19} ∪ {100, 101, . . .} ∪ · · · of all natural numbers whose base-10 representation begins with the digit 1 is not fractionally dense. This occurs despite the fact that d(A) = 19 , so that a positive proportion of the natural numbers belongs to A. The consideration of other bases reveals the following gem. http://dx.doi.org/10.4169/amer.math.monthly.121.07.590 MSC: Primary 11A99, Secondary 11B99; 11A16 590 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 This content downloaded from 134.173.131.217 on Tue, 17 Jan 2017 18:40:13 UTC All use subject to http://about.jstor.org/terms Gem 1. The set of all natural numbers whose base-b representation begins with the digit 1 is fractionally dense for b = 2, 3, 4, but not for b ≥ 5. To show this, we require the following more general result. Proposition 1. Let 1 < a ≤ b. The set

我们的目的是介绍有关这些观察的四个出色的事实。 {1，2，。。目前的四个出色的结果似乎在每月的每月忽略了，多年来的报价覆盖范围[3、4、5、9、10、11、14]。在其他地方出现，但以下几乎是未知的a的下部渐近密度是数量d（a）= lim in→∞| a（n）| n，它满足明显的边界0≤d（a）≤1。 r（a）中的r等于[0，∞）（即​​，如果r（a）在[0，∞）中的密集。表示始于数字1的b = 2、3、4，但对于b≥5。2。对于每个δ∈[0，12），而d（a）=则存在A集a n n n a。另一方面，如果d（a）≥1，则必须分馏，则必须将其分成三组。这样的分区仅使用两个集[2]。长度≥3。基本-B表示。 {100，101。}。自然数的比例属于A。其他基础的考虑揭示了以下宝石c©美国数学协会[每月121此内容从134.173.131.217下载，在周二，2017年1月17日在星期二18:40:13 UTC都使用http://about.jstor.jstor.org.org/terms gem 1。所有基本B表示的自然数均以数字1开始，对于B = 2、3、4，但对于B≥5的含量。 A≤b