Diophantine approximation in fields of characteristic 𝑝

Abstract

Listen

Translate article

where x is an indeterminate and the coefficients ci all belong to a fixed finite field GF(pn). We shall discuss a number of problems of diophantine approximation related to the numbers of 4(. We first (?3) prove an analogue of Kronecker's theorem; the theorem has been proved previously by MIahler [8, p. 514]. We next define uniform distribution of sequences of numbers in 4( (see [9], also [7, chap. 8]) and prove a number of theorems similar to the theorems of Weyl's well known paper. The sum S= Ee(k(A)), extended over polynomials A eGF[p n x] of degree less than m, where 0(u) is a polynomial of degree k and e(a) is defined in (2.3) below, is studied by Weyl's method of approximation. It is found that if at least one coefficient of ?(u) -0(0) is irrational and 1 coc. The case k > p is left open; there are polynomials of degree p for which S=pnm (see (6.8) and (6.9)). For k = 2, p > 2, we make a more detailed study of the sum