Complete Invariant of Integer Angles

Abstract

In this chapter we generalize the classical geometric interpretation of regular continued fractions presented in the previous chapter to the case of arbitrary integer angles, constructing a certain integer broken line called the sail of an angle. We combine the integer invariants of a sail into a sequence of positive integers called an LLS sequence. From one side, the notion of LLS sequence extends the notion of continued fraction, about which we will say more in the next chapter. From another side, LLS sequences distinguish the integer angles. Sails and LLS sequences of angles play a central role in the geometry of numbers. In particular, we use LLS sequences in integer trigonometry and its relations to toric singularities and in Gauss’s reduction theory. F. Klein generalized the notion of sail to the multidimensional case to study integer solutions of homogeneous decomposable forms. We will study this generalization in the second part of this book.

In this chapter we give definitions of an integer sines, sails of integer angles and corresponding LLS sequences. We prove that an LLS sequence is a complete invariant of an integer angle.