Weyl sums in with digital restrictions

Let be a finite field and consider the polynomial ring . Let . A function , where G is a group, is called strongly Q-additive, if f(AQ+B)=f(A)+f(B) holds for all polynomials with degBQ. We estimate Weyl sums in restricted by Q-additive functions. In particular, for a certain character E we study sums of the form

where is a polynomial with coefficients contained in the field of formal Laurent series over and the range of P is restricted by conditions on f_i(P), where f_i (1ir) are Q_i-additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form

with . Sums of this shape are treated by applying the kth iterate of the Weyl–van der Corput inequality and studying higher correlations of the functions f_i. With these Weyl sum estimates we show uniform distribution results.