Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 2 / e

Solution

Write

H = H (α)

. Since

α

has degree at least two and lies in

(0, 1)

H > 1

. Put

x = \frac{l o g ( nh )}{d l o g H} > 1, k = ⌊ x ⌋ + 1.

Then

H^{k d} > nh

and

k + 1 \leq x + 2 \leq 3 x .

Let

m = ⌈ \frac{n}{k + 1} ⌉ .

Part (d), applied with

m

in place of its polynomial-degree parameter, supplies

ℓ \in {k, k + 1}

such that no nonzero integer polynomial of degree at most

m - 1

h

vanishes at

β = α^{ℓ}

h^{m}

\sum_{j = 0}^{m - 1} a_{j} β^{j}, 0 \leq a_{j} < h,

are distinct. Since

(m - 1) ℓ \leq (m - 1) (k + 1) \leq n - 1,

they form a subset of

A_{n}

. Hence

∣ A_{n} ∣ \geq h^{m} \geq h^{n / (k + 1)} \geq h^{d n l o g H / (3 l o g (nh))} = H^{d n l o g h / (3 l o g (nh))},

as required.

Solved by gpt-5.6-sol high.

There are no discussions about this article yet.

There are currently no matching articles.