To print the names of all files in and below $dir of size 0:
find "$dir" -size 0
Note that not all implementations of find will produce output by default, so you may need to do:
find "$dir" -size 0 -print
Two comments on the final loop in the question:
Rather than iterating over every other word in a string and seeing if the alternate values are zero, you can partially eliminate the issue you're having with whitespace by iterating over lines. eg:
printf '1 f1\n0 f 2\n10 f3\n' | while read size path; do
test "$size" -eq 0 && echo "$path"; done
Note that this will fail in your case if any of the paths output by ls contain newlines, and this reinforces 2 points: don't parse ls, and have a sane naming policy that doesn't allow whitespace in paths.
Secondly, to output the data from the loop, there is no need to store the output in a variable just to echo it. If you simply let the loop write its output to stdout, you accomplish the same thing but avoid storing it.