On metrical theory of diophantine approximation over imaginary quadratic field

Metrical Diophantine approximation for quaternions

The metrical theory of Diophantine approximation for quaternions is developed using recent results in the general theory. In particular, Quaternionic analogues of the classical theorems of Khintchine, Jarnı́k and Jarnı́k-Besicovitch are established. Introduction Diophantine approximation begins with a more quantitative understanding of the density of the rationals Q in the reals R. The starting p...

A note on zero-one laws in metrical Diophantine approximation

q ψ(q) diverges but A(ψ) is of zero measure. In other words, without the monotonicity assumption, Khintchine’s theorem is false and the famous Duffin-Schaeffer conjecture provides the appropriate statement. The key difference is that in (1), we impose coprimality on the integers p and q. Let A(ψ) denote the resulting subset of A(ψ). The Duffin-Schaeffer conjecture states that the measure of A(ψ...

On a theorem of V. Bernik in the metrical theory of Diophantine approximation

Diophantine Equations and Class Numbers of Imaginary Quadratic Fields

Let A,D, K, k ∈ N with D square free and 2 | /k, B = 1, 2 or 4 and μi ∈ {−1, 1}(i = 1, 2), and let h(−21−eD)(e = 0 or 1) denote the class number of the imaginary quadratic field Q( √−21−eD). In this paper, we give the all-positive integer solutions of the Diophantine equation Ax +μ1B = K ( (Ay +μ2B)/K )n , 2 | / n, n > 1 and we prove that if D > 1, then h(−21−eD) ≡ 0(mod n), where D, and n sati...

Metric Diophantine Approximation over a Local Field of Positive Characteristic

We establish the conjectures of Sprindžhuk over a local field of positive characteristic. The method of KleinbockMargulis for the characteristic zero case is adapted.