### The Critical Lattices of a Circular Quadrilateral Formed by Arcs of Three Circles

Quarterly Journal of Mathematics, Vol. 17 (1946), 223-239

[Received 1 October 1945]

Introduction

Very recently I proved a result concerning lattice points in a non-convex region bounded by arcs of four circles. The proof provided an extremely simple example of a method due to Mordell; it was entirely self-contained, depending on no theory other than Minkowski's classical theorem on linear forms.

I now notice that similar results (from which the earlier results could in fact have been deduced) hold for another circular quadrilateral bounded by arcs of *three* circles. The proof again is very simple if use is made of some general properties of "star domains", i.e. regions which are symmetrical about the origin and in which every radius from the origin meets the boundary, a continuous curve, in just one point. The theory of plane star domains has been fully developed by Mahler. In the application to particular cases his methods may give rise to a large number of difficult extremal problems. It is therefore of some interest that these methods (which are essentially a development of those used by Minkowski for convex regions) should, in the present case, lead to a very simple proof indeed.

The region K_{r} considered here is bounded by arcs of three circles [±½], [r], defined by the equations

x^{2} + y^{2} = ±x, x^{2} + y^{2} = r^{2} (0 < r < 1).

More precisely, if P is any point of K_{r}, then its coordinates (x,y) satisfy at least one of the inequalities

x^{2} + y^{2} ≤ |x|,

x^{2} + y^{2} ≤ r^{2} (0 < r < 1),

i.e. K_{r} is a closed star domain and consists of the points (x,y) satisfying

min{(x^{2}+y^{2})^{2}/x^{2}, (x^{2}+y^{2})/r^{2}} ≤ 1,

where the expression in the bracket is homogeneous.