Is there a name for the set of real numbers which can be written as the sum of finitely many powers of 2?
Such a set would include all integers of course, but no irrational and not all rational numbers (1/2 is in, but 1/3 is not for example).
Does this set have any interesting properties or uses? Is it well defined?
These are the dyadic rationals (or the positive dyadic rationals, if you don't allow subtractions as well as sums).
This isn't hard to show. First, suppose $q$ is a dyadic rational. let $a$ be odd and $b=2^m$ be a power of $2$ with ${a\over b}=q$; then $q=2^{-m}+2^{-m}+...$ ($a$ times).
In the other direction, suppose I have $q=2^{i_1}+2^{i_2}+...+2^{i_n}$; I want to show that that's a dyadic rational. WLOG, suppose $i_1\le i_2\le . . . \le i_n$. Then we can factor $q$ as $$2^{i_1}(1+2^{j_2}+2^{j_3}+...+2^{j_n}),$$ where $j_k=i_k-i_1$. Now since $i_1\le i_k$ for all $k$, we have that $(1+2^{j_2}+2^{j_3}+...+2^{j_n})$ is an integer; call it "$c$".
So $q=2^{i_1}c$. If $i_1\ge 0$, then $q$ is an integer - certainly a dyadic rational! If $i_1<0$, then $q={c\over 2^{-i_1}}$, so is a dyadic rational.