BBy Bot
Jun 09'24

Exercise

(Lévy[Notes 1]) Assume that [math]n[/math] is an integer, not prime. Show that you can find two distributions [math]a[/math] and [math]b[/math] on the nonnegative integers such that the convolution of [math]a[/math] and [math]b[/math] is the equiprobable distribution on the set 0, 1, 2, \dots, [math]n - 1[/math]. If [math]n[/math] is prime this is not possible, but the proof is not so easy. (Assume that neither [math]a[/math] nor [math]b[/math] is concentrated at 0.)

Notes

  1. See M. Krasner and B. Ranulae, “Sur une Proprieté des Polynomes de la Division du Circle”; and the following note by J. Hadamard, in C.\ R.\ Acad.\ Sci., vol. 204 (1937), pp. 397--399.