  Given a text $T$ of length $n$ and a pattern $P$ of length $m$, the
approximate pattern matching problem asks for computation of a particular
\emph{distance} function between $P$ and every $m$-substring of $T$. We
consider a $(1\pm\varepsilon)$ multiplicative approximation variant of this
problem, for $\ell_p$ distance function. In this paper, we describe two
$(1+\varepsilon)$-approximate algorithms with a runtime of
$\widetilde{O}(\frac{n}{\varepsilon})$ for all (constant) non-negative values
of $p$. For constant $p \ge 1$ we show a deterministic
$(1+\varepsilon)$-approximation algorithm. Previously, such run time was known
only for the case of $\ell_1$ distance, by Gawrychowski and Uzna\'nski [ICALP
2018] and only with a randomized algorithm. For constant $0 \le p \le 1$ we
show a randomized algorithm for the $\ell_p$, thereby providing a smooth
tradeoff between algorithms of Kopelowitz and Porat [FOCS~2015, SOSA~2018] for
Hamming distance (case of $p=0$) and of Gawrychowski and Uzna\'nski for
$\ell_1$ distance.
