  According to a classical result of Szemer\'{e}di, every dense subset of
$1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large
enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson
says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large
grid, if $N$ is large enough. Here we generalize these results for separated
point sets on the line and respectively in the Euclidean space: (i) every dense
separated set of points in some interval $[0,L]$ on the line contains an
arbitrary long approximate arithmetic progression, if $L$ is large enough. (ii)
every dense separated set of points in the $d$-dimensional cube $[0,L]^d$ in
$\RR^d$ contains an arbitrary large approximate grid, if $L$ is large enough. A
further generalization for any finite pattern in $\RR^d$ is also established.
The separation condition is shown to be necessary for such results to hold. In
the end we show that every sufficiently large point set in $\RR^d$ contains an
arbitrarily large subset of almost collinear points. No separation condition is
needed in this case.
