I try to find sequences of digits (in base $b$, with $b$ not necessarily an integer) that are not cross-correlated. While the digits in base $b$ of (say) $\pi$ and $e$ do not exhibit auto-correlations when taken separately (assuming $b$ is an integer) since these numbers are believed to be normal numbers, what about cross-correlations between these two sequences of digits?
The context is a business application: a generic number guessing game played with real money. If I use sequences that are cross-correlated, the player can leverage this fact (if she discovers the auto-correlations) to increase her odds of winning, making the game unfair to the operator. In short, I could lose money. For details, see section 4 in my article Some Fun with Gentle Chaos, the Golden Ratio, and Stochastic Number Theory.
For those interested, the picture below shows the lag-1 auto-correlation of digits in base $b$ for normal numbers such as $\pi, e, \sqrt{2}, \log 2$ and so on. The auto-correlation is zero if and only if $b$ is an integer, but it is as low as $(-3 + \sqrt{5})/2$ (this is the global minimum) when $b=(1+\sqrt{5})/2$. Interestingly, the lag-$k$ auto-correlation in base $b$ is equal to the lag-1 auto-correlation in base $b^k$.
