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, 5, 4, 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, 1 2 ), there exists a set A ⊂ N with d(A) = δ that is not fractionally dense. On the other hand, if d(A) ≥ 1 2 , then A must be fractionally dense [15]. 3. One 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 which 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) = 1 9 , so that a positive proportion of the natural numbers belongs to A. The consideration of other bases reveals the following gem. 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. January 2014] FOUR QUOTIENT SET GEMS 1 Mathematical Assoc. of America American Mathematical Monthly 121:1 December 5, 2013 1:25 a.m. 4QSG.tex page 2 To show this, we require the following more general result. Proposition 1. Let 1 < a ≤ b. The set