Energy-efficient threshold circuits computing MOD functions

We prove that the modulus function MOD_m of n variables can be computed by a threshold circuit C of energy e and size s = O(e(n/m) ^{1/(e - 1)}) for any integer e ≥ 2, where the energy e is defined to be the maximum number of gates outputting "1" over all inputs to C, and the size s to be the number of gates in C. Our upper bound on the size s almost matches the known lower bound s = Ω(e(n/m)^1/e). We also consider an extreme case where threshold circuits have energy 1, and prove that such circuits need at least 2^{(n - m)/2} gates to compute MOD _m of n variables.