A divergent Khintchine theorem in the real, complex, and p-adic fields

Abstract

In this paper, we show that if the sum ∞ r=1 Ψ (r) diverges, then the set of points (x, z, w) ∈ R × C × Q p satisfying the inequalities |P (x)| < H −v1 Ψ λ1 (H), |P (z)| < H −v2 Ψ λ2 (H), and |P (w)| p < H −v3 Ψ λ3 (H) for infinitely many integer polynomials P has full measure. With a special choice of parameters v i and λ i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.

Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

“…Similarly to what Beresnevich, Bernik and Götze did in [6], we call a tailored polynomial an irreducible polynomial which satisfies (11). Our construction follows closely the argument of [6, Section 3], and it is based on Theorem 2.7, which we will then prove in Section 8 using the quantitative non-divergence method of Kleinbock and Margulis.…”

“…and this was later proved by Bernik in [9]. The problem (2) was then considered for abitrary ψ and m = 1 in [10], as well as for the case where the variables x k can also take complex or p-adic values in [11,15] (m = 2) and [20,19] (arbitrary m). In particular, the following result is contained in the preprint [5], which deals with the more general case of systems of linear forms in dependent variables, i.e.…”

Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables

“…The next major step was generalisation of the Želudevič's results to systems of inequalities where the right-hand sides are arbitrary functions ψ H ( ). Analogues of Khintchine's theorem were proved for both convergence and divergence cases [78][79][80].…”

Problem of Nesterenko and Method of Bernik

“…Knowledge of the nature of the roots is very important in the problems of Diophantine approximations for construction of regular systems [3,4]. Numerous applications of this concept arose when obtaining estimates for the Hausdorff measure and Hausdorff dimension of Diophantine sets [5] and proving analogues of the Khintchine theorem [6,7]. Using the regular systems, the exact theorems on approximation of real numbers by real algebraic [6], by algebraic integers [8], of complex numbers by complex algebraic [9] were obtained, and similar problems in the field of -adic numbers [10] and in R × C × Q [7] were investigated.…”