Knowlpedia

Theorem

Quadratic irrational lower bound

Integer linear forms in a fixed quadratic irrational have a reciprocal polynomial lower bound.

Statement

For a fixed α\alpha, there is a constant cα>0c_\alpha>0 such that

∣m+nα∣≥cα1+m2+n2for all (m,n)∈Z2∖{(0,0)}.|m+n\alpha|\ge\frac{c_\alpha}{1+\sqrt{m^2+n^2}} \quad\text{for all }(m,n)\in\mathbb Z^2\setminus\{(0,0)\}.
Proof

With a,b,ca,b,c and α′\alpha' as in the conjugate identity, the product a(m+nα)(m+nα′)a(m+n\alpha)(m+n\alpha') is a nonzero integer and has absolute value at least one. Also ∣m+nα′∣≤Cαm2+n2|m+n\alpha'|\le C_\alpha\sqrt{m^2+n^2}. Dividing gives the claimed bound.

Rational approximation

There is consequently c′ ⁣>0c'\!>0 such that ∣α−p/q∣≥c′/q2|\alpha-p/q|\ge c'/q^2 for all integers pp and q≥1q\ge1. When ∣p/q−α∣≤1|p/q-\alpha|\le1, apply the linear-form estimate and use ∣p∣≤(∣α∣+1)q|p|\le(|\alpha|+1)q; when the difference exceeds one, reduce the constant. This is the badly approximable property of quadratic irrationals.

References