  We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots,
x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If
lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other
than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large
enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from
an interval, the length of which depends only on $n$, $p$, $d$, and the measure
of $A$. Known counterexamples prevent generalizations of this result to the
remaining values of the exponent $p$. We also give an explicit bound for the
length of the aforementioned interval. The proof makes the bound depend on the
currently available bounds in Szemer\'{e}di's theorem on the integers, which
are used as a black box. A key ingredient of the proof are power-type
cancellation estimates for operators resembling the multilinear Hilbert
transforms. As a byproduct of the approach we obtain a quantitative improvement
of the corresponding (previously known) result for side lengths of
$n$-dimensional cubes with vertices lying in a positive measure subset of
$([0,1]^2)^n$.
