For the counting-measure <Lp norm>, define
$$
\operatorname{AP}_{2m}(f;X)
=\{t\in\mathbb F_p^n:\|\tau_tf-f\|_{2m}\leq X\},
\qquad (\tau_tf)(x)=f(x+t).
$$
This is the set of <Lp almost periods> of $f$ with exponent $p=2m$ and error $X$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 3 / a / Solution

{title2=$L^p$ almost period}

For $f:G\to\mathbb C$ and $X\geq0$, an $L^p$ almost period of $f$ with error $X$ is an element $t\in G$ such that
$$
\|\tau_tf-f\|_p\leq X,
\qquad (\tau_tf)(x)=f(x+t).
$$


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