Linear Systems over Composite Moduli

We study solution sets to systems of ´generalized´ linear equations of the form: ¿_i (x₁, x₂, ···, x_n) in ¿ A_i (mod m) where ¿₁,..., ¿_t are linear forms in n Boolean variables, each A_i is an arbitrary subset of Z_m, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. with the boolean function MOD_q, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ o AND o MOD^A _m (i.e having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MOD_m gates at the base) computing the function MOD_q. This settles an open problem of Beigel and Maciel (Complexity´97) for the case of such modulus m. Our technique makes use of the work of Bourgain on estimating exponential sums involving a low-degree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov on arithmetic circuits over finite fields.