  Furstenberg, Katznelson and Weiss proved in the early 1980s that every
measurable subset of the plane with positive density at infinity has the
property that all sufficiently large real numbers are realised as the Euclidean
distance between points in that set. Their proof used ergodic theory to study
translations on a space of Lipschitz functions corresponding to closed subsets
of the plane, combined with a measure-theoretical argument. We consider an
alternative dynamical approach in which the phase space is given by the set of
measurable functions from $\mathbb{R}^d$ to $[0,1]$, which we view as a compact
subspace of $L^\infty(\mathbb{R}^d)$ in the weak-* topology. The pointwise
ergodic theorem for $\mathbb{R}^d$-actions implies that with respect to any
translation-invariant measure on this space, almost every function is
asymptotically close to a constant function at large scales. This observation
leads to a general sufficient condition for a configuration to occur in every
set of positive upper Banach density at all sufficiently large scales,
extending a recent theorem of B. Bukh. To illustrate the use of this criterion
we apply it to prove a new result concerning three-point configurations in
measurable subsets of the plane which form the vertices of a triangle with
specified area and side length, yielding a new proof of a result related to
work of R. Graham.
