What is a sparse set? 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?

According to a short monograph entitled _Sur les Ensembles Clairsemés_ published in French by Z. Semadeni in 1959 (freely available at < 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).