I read a question that asked "prove that the set of all positive integers expressible as the sum of two integers square has zero density." Now I was under the impression that something was dense iff any subset nonempty subset of the reals contained one of those numbers. For example, since and nonemty subset of the reals contains a rational, we can say the rationals are dense over the reals. So what does zero density mean?
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1$\begingroup$ I'm closing this because the answer is very easy to find using the canonical search - en.wikipedia.org/wiki/Density_(disambiguation) $\endgroup$– François G. DoraisCommented Sep 1, 2011 at 23:55
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1$\begingroup$ Your interpretation of density is not correct -- $\{\sqrt{2}\}$ is a non-empty subset of the reals which contains no rationals. If you read the question as part of a course, then hopefully it should have been made clear earlier what definition of density is being used (see the link in David White's answer). $\endgroup$– Yemon ChoiCommented Sep 1, 2011 at 23:57
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There are several different notions of density in the integers. The most common is Natural Density. You can learn more at wikipedia. The idea is that you want to measure how large a subset of the integers is. One attempt is via measure theory. It turns out measure theory often works best in continuous settings, while the integers are discrete. For instance, you can't use the standard measure on $\mathbb{R}$ because the integers already have measure zero. There are plenty of other measures you can define on $\mathbb{Z}$ but it's not clear which is the best to capture number theoretic sizes like the "size" of the squares vs. all integers. Another attempt to measure size is via Baire Category, and this works well in general topological spaces. Again, we're not picking up the number-theoretic information in the integers, and that's why natural density comes into play.
answered Sep 1, 2011 at 23:55