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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 129

Put $f=1_{-A}*\mu_A$. The probability measure
$$
\mu=\mu_Y^{(k)}*\mu_{-Y}^{(k)}
$$
is supported on $kY-kY$. Convolution by $\mu$ averages translates by points in this support, so convexity of the supremum norm gives
$$
\|f*\mu-f\|_\infty
\le\mathbb E_{z\sim\mu}\|\tau_zf-f\|_\infty
\le\epsilon.
$$
Thus
$$
\boxed{\|1_{-A}*\mu_A*\mu-1_{-A}*\mu_A\|_\infty\le\epsilon.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / iii / Solution

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