I see this as more of a set selection problem or a combinatoriall design problem than a number theory problem, unless there is something else about the $a_{ijkl}$ that is not being mentioned.

By setting 4 of the w's to 1show and the rest to 0, I can convince myself that 5is of the a's are zero. Conversely, choose the a's anyway you like, but let me set 5selection of them to zero: $a_{1111}$, and four others which partition the index set. Then for any tuple of w's, either one of the four other a's I picked selects a below average subtuple of w's to sum, or they are all above average and w_1 is less than 1/17th of the sum. Your inequalitty will be satisfied by this selection of a's and any w's, and it will not tell me a thing about the a's you picked.

Gerhard "Ask Me About System Design" Paseman, 2011.09.06

I see this as more of a set selection problem or a combinatorial design problem than a number theory problem, unless there is something else about the $a_{ijkl}$ that is not being mentioned.

By setting 4 of the w's to 1 and the rest to 0, I can convince myself that 5 of the a's are zero. Conversely, choose the a's anyway you like, but let me set 5 of them to zero: $a_{1111}$, and four others which partition the index set. Then for any tuple of w's, either one of the four other a's I picked selects a below average subtuple of w's to sum, or they are all above average and $w_1$ is less than 1/17th of the sum. Your inequality will be satisfied by this selection of a's and any w's, and it will not tell me a thing about the a's you picked.

Gerhard "Ask Me About System Design" Paseman, 2011.09.06