Roots of Markoff quadratic forms as strongly badly approximable numbers

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For a real number x, x = {|x-p|: p∈ Z} is the distance of x to the nearest integer. We say that two real numbers θ, θ' are ± equivalent if their sum or difference is an integer. Let θ be irrational and put ϕ(θ) = {q q θ: q ∈ N }. We will prove: If ϕ(θ)> 1/3, then θ is ± equivalent to a root of fₘ (x,1) = 0, where fₘ is a Markoff form. Conversely, if θ is ± equivalent to a root of fₘ(x,1)=0, then ϕ(θ) = m mθ = 23+√9-4m⁻² > 1/3.