Consider \sum_n x_n a_n, the summation of the weighted Rademacher series, where the Rademacher series (x_n) is a binary sequence of -1 and 1, and the weight (a_n) is a sequence of real numbers. The Rademacher theorem states that the summation diverges for almost every sequence (x_n) when \sum_n a^2_n is infinite and converges for almost every (x_n) when \sum_n a^2_n is finite. The theorem gives a nice result about the amount of (x_n) that makes the summation diverge/converge and how this depends on the weight (a_n). However, it says nothing about the structure of the desired (x_n).

In this talk, we will use the language of algorithmic randomness to give a more precise description of the theorem. We will study the class of desired (x_n) and compare it with some common concepts in computability theory.