2021/12/27 by Mosunov, Anton
#11D59 #11J61 #11J68 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2112.13919
Let α be a real algebraic number of degree d ≥ 3 and let β∈ \mathbb Q(α) be irrational. Let μ be a real number such that (d/2) + 1 < μ< d and let C0 be a positive real number. We prove that there exist positive real numbers C1 and C2, which depend only on α, β, μ and C0, with the following property. If x1/y1 and x2/y2 are rational numbers in lowest terms such that H(x2, y2) ≥ H(x1, y1) ≥ C1 and |α- (x1)/(y1)| lt; (C0)/(H(x1, y1)μ), |β- (x2)/(y2)| lt; (C0)/(H(x2, y2)μ), then either H(x2, y2) > C2-1 H(x1, y1)μ- d/2, or there exist integers s, t, u, v, with sv - tu ≠ 0, such that β= (sα+ t)/(uα+ v) and (x2)/(y2) = (sx1 + ty1)/(ux1 + vy1), or both. Here H(x, y) = max(|x|, |y|) is the height of x/y. Since μ- d/2 exceeds one, our result demonstrates that, unless α and β are connected by means of a linear fractional transformation with integer coefficients, the heights of x1/y1 and x2/y2 have to be exponentially far apart from each other. An analogous result is established in the case when α and β are p-adic algebraic numbers.