Strong Normality, Modular Normality, and Flat Polynomials: Applications of Probability in Number Theory and Analysis

We use probabilistic methods, along with other techniques, to address three topics in number theory and analysis. Champernowne's number is well known to be normal, but the digits are highly patterned. The definition of normality reflects the convergence in frequency of the digits of a random number, but the behaviour of the discrepancy is better described by the law of the iterated logarithm. We use this to define "strong normality," and find that almost all numbers are strongly normal, and strongly normal numbers are normal. However, the base-2 Champernowne number is not strongly normal in the base 2. We use a method of Sierpinski to construct a number strongly normal in every base. Next, we define normality of an integer sequence modulo an integer q; this is a refinement of the existing notion of uniform distribution modulo q. If alpha is normal in the base r, the sequence given by the integer part of r^n alpha is uniformly distributed modulo every integer q > 1; however, the sequence is normal modulo q if and only if q divides r. This particular sequence does show pseudorandom behaviour modulo every q > r; we define "base-r normality modulo q" to capture this behaviour. The third topic concerns flat polynomials. A sequence of polynomials is "flat" if its values on the unit circle are bounded above and below by absolute constant multiples of p^n, where n is the degree. Beck showed that there exist flat sequences of polynomials with coefficients that are l-th roots of unity, for every l greater than some lower bound. Beck gave a lower bound of 400, but we correct a minor error in his proof and show that this should have been 851. Beck relied on a constant from Spencer's work on the discrepancy of linear forms. We repeat Spencer's calculation, slightly improving the value of his constant and giving a new bound of 492. An improvement of Spencer's method, due to Kai-Uwe Schmidt, allows us to lower the bound to 345.