Simultaneous Diophantine approximation in the real, complex and p–adic fields.

Abstract

In this paper it is shown that if the volume sum ∑r = 1∞ Ψ(r) converges for a monotonic function Ψ then the set of points (x, z, w) ∈ ℝ × ℂ × ℚp which simultaneously satisfy the inequalities |P(x)| ≤ H−v1 Ψλ1(H), |P(z)| ≤ H−v2 Ψλ2(H) and |P(w)|p ≤ H−v3 Ψλ3(H) with v1 + 2v2 + v3 = n − 3 and λ1 + 2λ2 + λ3 = 1 for infinitely many integer polynomials P has measure zero.

SIMULTANEOUS DIOPHANTINE APPROXIMATION IN R2 × C × Qp

“…We use scheme of the proofs of propositions 1, 2, 3 4 of [3], correspondingly but there exist some distinctions. The distinctions appear in the sets of X = (X 1 , X 2 , X 3 , X 4 ) of the corresponding spaces.…”

“…The distinctions appear in the sets of X = (X 1 , X 2 , X 3 , X 4 ) of the corresponding spaces. Namely, in [3] we have X = (…”

“…(1) we use Lemmas 1-3 §3; in (2) we use Lemmas 1-4 §3 and make a reduction to polynomials of the third degree (in [3] the reduced polynomials have the second degree); in…”

“…we use Lemmas 1-4 §3 and make a reduction to polynomials of the forth degree (in [3] the ones have the third degree); in (4) we use Lemmas 1-3 §3 and make a reduction to polynomials of the third degree (in [3] the ones have the second degree).…”

“…It is a combination of Propositions 6, 7 [3], where for the second coordinate i 2 (i 2 = 1) we add the inequality q 21 + k 22 /T ≥ 1 + v 2 + λ 2 , and for the third coordinate i 3 (i 3 = 1) we take r 1 + l 2 /T ≥ 1 + v 3 + λ 3 .…”