16 - More about conditional Cauchy equations. Applications to additive number theoretical functions and to coding theory

Summary

This chapter is entirely devoted to conditional Cauchy equations which were introduced in Chapter 6 and already studied in Chapter 7. In the first section, by using a fixed point theorem and also the order structure of ℝ, and by adding a strong regularity condition on the unknown function, we deduce the addundancy of some conditions on rather thin sets. In the second and third sections, we use group decomposition and then proceed to an application to additive number theory. An analogous equation in the fourth section yields an application to information theory, for mean codeword lengths. In the two last sections, we come back to additive number theory, proving the basic result that logarithms are the only monotonic functions defined on integers which transform a product into a sum. We extend this and similar results in various ways.

Expansions of the Cauchy equation from curves

Thus far, for conditional Cauchy equations, we have been dealing with rather ‘large’ subsets Z (see Chapters 6 and 7). Under some additional regularity assumptions on f, we may proceed to far smaller Z. A typical result can be obtained in the plane by using for Z some curve (Dhombres 1979, pp. 3.32–3.37). We recall from Section 7.1 that Z ⊂ ℝ² is addundant in a class of functions if the restriction of the Cauchy equation from ℝ² to Z changes neither the domain nor the form of its general solution in the class.