Combinatorics of the Gauss digitization under translation in 2D

International audienceThe action of a translation on a continuous object before its digitization generates several digital objects. This paper focuses on the combinatorics of the generated digital objects up to integer translations. In the general case, a worst-case upper bound is exhibited and proved to be reached on an example. Another upper bound is also proposed by making a link between the number of the digital objects and the boundary curve, through its self-intersections on the torus. An upper bound, quadratic in digital perimeter, is then derived in the convex case and eventually an asymptotic upper bound, quadratic in the grid resolution, is exhibited in the polygonal case. A few signicant examples finish the paper

The number of configurations in lattice point counting I

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

Analytic Number Theory

[no abstract available

Problems related to lattice points in the plane

In the first part of this research we find an improvement to Huxley and Konyagin's current lower bound for the number of circles passing through five integer points. The improved lower bound is the conjectured asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem. In the second part of the research we consider questions linked to the distribution of different configurations of integer points of the circle passing through the unit square. We show that different configurations of points are distributed uniformly throughout the unit square for circles of fixed radius. Results are obtained by looking at the distribution of the crossing points of circles, where the circles form the boundaries of domains. The domain of a configuration is the set of possible positions of the centre of the circle within the configuration. We choose a rectangle within the unit square and then count the number of regions of the rectangle which are formed by domain boundaries

The number of N point digital discs

A digital disc is the set of all integer points inside some given disc. Let {\cal D}_{N} be the number of different digital discs consisting of N points (different up to translation). The upper bound {\cal D}_{N} = {\cal O}(N^{2}) was shown recently; no corresponding lower bound is known. In this paper, we refine the upper bound to {\cal D}_{N} = {\cal O}(N), which seems to be the true order of magnitude, and we show that the average \overline{\cal D}_{N} = \left({\cal D}_{1} + {\cal D}_{2} + \ldots + {\cal D}_{N}\right)/N has upper and lower bounds which are of polynomial growth in N