Exercises: Almost linear, Almost exponential

When I began actively pursuing the application of approximate functional equations to number theory, in the early seventies, results of the Ulam–Hyers type were sparse. Moreover, they did not lend themselves to the problems which I had to hand. It should be emphasised that the method of the stable dual is not concerned with the approximate functional equations that arise, for example, in the theory of the Riemann zeta function. In that theory approximate functional equations are established for certain given functions, mainly sums of exponentials. In a sense an analytic reciprocity law is derived. In the method of the stable dual an unknown function is assumed to satisfy a weak global constraint, and as far as possible the local nature of the function is then determined. As applied to number theory the method of the stable dual typically gives rise to a complicated approximate functional equation involving several functions and many variables. The first step is to tease out an approximate equation of a more manageable type. This step depends upon the number theoretic and distributional properties of the objects under consideration. The appropriate notion of stability is then determined by the number theoretic application in view. My aim was usually towards an equation with continuous rather than discrete variables. Although by 1980 I had developed a tolerable technique for treating approximate functional equations arising in the study of arithmetic functions, I felt the need to better understand some of the arguments.