Inhomogeneous Non-linear Diophantine Approximation

Let be a strictly positive monotonically decreasing function deened on the set of positive integers. Given real numbers and , consider the solubility of the following two inequalities jq + pj < (q); (1) jq + p + j < (q) (2) for integers p and q. The rst problem is said to be homogeneous and the second inho-mogeneous (see 2]). The well known theorem of Khintchine 2, 4] asserts that for almost all 2 R in the sense of the Lebesgue measure, the inequality (1) has only nitely many solutions if the series 1 X k=1 (k) (3) converges (in which case need not be monotonic) and has innnitely many solutions if the series diverges. An analogous result is true for the inhomogeneous problem: given any real number , the inequality (2) holds for innnitely many q for almost no when the sum (3) converges and for almost all when the sum (3) diverges 6]. Thèdoubly metric case' in which one considersàlmost all' as well, is easier and holds without monotonicity (see 2, Chapter 7]). Schmidt's generalization 6] of the higher dimensional Khintchine-Groshev Theorem 7, pp. 28,33] includes the inhomogeneous case and implies that given y 2 R, the inequality ja n x n + + a 1 x + a 0 ? yj < H ?n+1 (H); (4) where H = maxfja n j; : : :; ja 0 jg, has only nitely or innnitely many integer solutions a 0 ; : : : ; a n for almost all x = (x 1 ; : : :x n) 2 R n according as the series (3) converges or diverges (the notation here diiers a little from that of 7]).