This [question][1] seems related to the case $N=2.$ I gave an answer suggesting that the continued fraction expansion of ${|e_2|}/{|e_1|}$ was the key. I haven't rechecked it.
I'd venture that already the case $N=3$ would be much more difficult.


  [1]: http://mathoverflow.net/questions/121826/on-the-set-of-numbers-generated-by-integer-linear-combination-of-two-real-number/121883#121883