Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

Abstract

The units in cubic number fields together with the uniform distribution theorem are used to prove the following theorem. Let 1, β 1, β 2 be a basis for a real cubic number field. Let C > 0 be a given constant. Let λ B equal the number of solutions in integers q, p 1, p 2 of the inequalities 0 < qβ i − p i < C q 1 2 ( i = 1, 2), 1 ≤ q ≤ B. Then λ B = O(1) or there is a constant C′ > 0 such that λ B ∼ C′ log B ( B → ∞). The stumbling blocks for generalizing this result to higher dimensions are discussed.