## Abstract

When a strictly convex plane set *S* moves by translation, the set *J* of points of the integer lattice that lie in *S* changes. The number *K* of equivalence classes of sets *J* under lattice translations (configurations) is bounded in terms of the area of the Brunn-Minkowski difference set of *S*. If *S* satisfies the Triangle Condition, that no translate of *S* has three distinct lattice points in the boundary, then *K* is asymptotically equal to the area of the difference set, with an error term like that in the corresponding lattice point problem. If *S* satisfies a Smoothness Condition but not the Triangle Condition, then we obtain a lower bound for *K*, but not of the right order of magnitude.

The case when *S* is a circle was treated in our earlier paper by a more complicated method. The Triangle Condition was removed by considerations of norms of Gaussian integers, which are special to the circle.

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