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In its article on thick sets, Wikipedia references the notion of a sparse set, but Wikipedia does not have an article on sparse sets. It doesn’t seem to be defined in Encyclopedia of Mathematics, nor in Wolfram MathWorld, either. I know that terms like ‘sparse vector’, ‘sparse matrix’, and even ‘sparse subset’ are out there, but what is a sparse set of real numbers? – And is the definition of a sparse set of positive integers expressible more simply than a sparse set of arbitrary real numbers?

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I don't think there is a precise definition of sparse sets. However, I think it makes sense to use the word sparse in that context. If you let $A_n = \{x \mid x = 10^n + m, 0\leq m \leq n\}$ note that \begin{align*} A_0 & = \{1\} \\ A_1 & = \{10, 11\} \\ A_2 & = \{100, 101, 102\}\\ A_3 & = \{1000, 1001, 1002, 1003\}\\ A_4 & = \{10000, 10001,10002,10003,10004\}\\ & \hspace{0.1 in}\vdots \end{align*}

In this sense $A_i$'s are very disjoint as $i \rightarrow \infty$.

user 1987
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According to a short monograph entitled Sur les Ensembles Clairsemés published in French by Z. Semadeni in 1959 (freely available at http://eudml.org/doc/268570), a set $A$ in a metric space is sparse if and only if it contains no nonempty, dense-in-itself subset (equivalently, $A$ is not dense in any perfect set).

mathematrucker
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    Thanks. This is why we need Esperanto. I've up-voted your answer, and will accept it. –  Aug 26 '17 at 15:08
  • Considering that Semadeni's paper is "cited by 45" in Google Scholar the checkmark seems justified despite the apparent nonexistence of any other references that use the term. Thanks for both that, and for mentioning Esperanto (which I had never heard of...Wikipedia page here). – mathematrucker Aug 26 '17 at 17:56
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    I happened to see this answer just now. The term "sparse" has no generally accepted meaning in mathematics and different groups of mathematicians have used it for different notions. What you cited is more commonly known as a scattered set. Another use of "sparse" was by Henry Blumberg (1930s and 1940s) and taken up by a few others, mostly (in the past few decades) by Ludek Zajicek. See here and here and here (pp. 75 on). – Dave L. Renfro Jan 03 '18 at 18:44