Breadcrumb
Almost modular functions and distribution modulo one
Supervisor: Jens Marklof
Theme: Number Theory
I have recently discovered a novel class of functions, the almost modular functions, or AMFs. Classical modular functions are among the most important objects of study in number theory. They are defined as functions that are invariant under the action of the modular group or one of its congruence subgroups. On the other hand, an almost modular function is a function which is approximately invariant, in that the error in the approximation tends to zero as one considers modular subgroups of increasing index. The main outcome of my investigation is that AMFs
satisfy very similar limit theorems as modular functions. An important example of an AMF is the error term for distribution of fractional parts of the well studied number-theoretic sequence
x, 4x, 9x, 16x,... It thus follows that the error term of this classical sequence satisfies a limit theorem, where the limiting distribution, perhaps surprisingly, deviates from a Gaussian law. There is a wealth of other examples of almost modular functions. The research on AMFs is only at its beginning!