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.Comment: 11 pages, 1 figure