 Approximation of Teichmuller lifts by iterated pth powers

ID: approximation-of-teichmuller-lifts-by-iterated-pth-powers

Codex  0 Created 2026-10-05 Updated 2026-10-07

For

K / Q_{p}

finite, let

x_{n}^{p^{n}} = x

in its finite residue field, and choose arbitrary lifts

y_{n} \in O_{K}

. With normalized discrete valuation,

v_{K} (u^{p} - v^{p}) \geq min {v_{K} (p) + r, p r} \geq r + 1 if v_{K} (u - v) \geq r \geq 1.

Applying this binomial estimate repeatedly to

y_{n}

and

[x_{n}]

proves the displayed bound, and hence convergence to

[x]

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