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    $\begingroup$ Your conjectured result, if true, would be up to $O(1/\epsilon)$, no? $\endgroup$ Commented Jul 14, 2020 at 17:51
  • $\begingroup$ Yes, unfortunately the closer to zero you get, my obvious guess is that the exception set (numbers that can not be represented that way) even though finite, would be extremely large. $\endgroup$ Commented Jul 14, 2020 at 18:00
  • $\begingroup$ A first step to prove conjecture A might be this: prove that if $S$ is equidistributed, then $S+S=\{x,y$ with $x,y \in S\}$ is also equidistributed. $\endgroup$ Commented Jul 14, 2020 at 18:02