For

M = max {∣ s ∣, ∣ t ∣, 1}

∣ s^{j} - t^{j} ∣ = ∣ s - t ∣ \sum_{l = 0}^{j - 1} s^{j - 1 - l} t^{l} \leq j M^{j - 1} ∣ s - t ∣.

Consequently the polynomial

f

∣ f (s) - f (t) ∣ \leq A M^{d - 1} ∣ s - t ∣, A = j = 1 \sum d j ∣ α_{j} ∣.

ii

Solution

Since

p / q

is not a root,

q^{d} f (p / q)

is a nonzero integer, and hence

∣ f (p / q) ∣ \geq q^{- d} .

Moreover

0 < θ - p / q < 1

implies

∣ p / q ∣ < ∣ θ ∣ + 1

. Applying part (i) to

s = θ

t = p / q

, and using

f (θ) = 0

gives

q^{- d} \leq A (∣ θ ∣ + 1)^{d - 1} θ - \frac{p}{q} .

Rearrangement proves

θ - \frac{p}{q} \geq \frac{1}{A} (\frac{1}{∣ θ ∣ + 1})^{d - 1} \frac{1}{q ^{d}} .

This is the core estimate behind the Liouville approximation theorem.

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