Weyl sums in Fq[x] with digital restrictions

Let Fq be a finite field and consider the polynomial ring Fq[X]. Let Q Fq[X]. A function f : Fq[X] G, where G is a group, is called strongly Q-additive, if f(AQ + B) = f(A) + f(B) holds for all polynomials A, B Fq[X] with deg B < deg Q. We estimate Weyl Sums in Fq[X] restricted by Q-additive functions. In particular, for a certain character E we study sums of the form P E(h(P)), where h Fq((X-1))[Y ] is a polynomial with coefficients contained in the field of formal Laurent series over Fq and the range of P is restricted by conditions on fi(P), where fi (1 i r) are Qi-additive functions. Adopting an idea of Gelfond such sums can be rewritten as sums of the form deg P <n E h(P) + r i=1 Ri Mi fi(A) , with Ri, Mi Fq[X]. Sums of this shape are treated by applying the k-th iterate of the Weylvan der Corput inequality and studying higher correlations of the functions fi. With these Weyl Sum estimates we show uniform distribution results.

• Type: Journal
• Year: 2008
• Where: FFA
• Authors: Manfred G. Madritsch, Jörg M. Thuswaldner
• Keywords: FFA 2008, Polynomial, Sums, Weyl Sum