Could it be that on every 53th position in the Pi sequence we will find abnormal many "7"? Or more general: does there exist a favourite distance S when stepping thru the sequence with equal steps of length S will find "much more" of digit X than of others?
To answer this we have to follow this certain number and analyze the position p of its appearences by the formula F = p mod k. In the example above k=53 and F will last from 0 to 52.
Performing this analysis for all ten digits from 0 to 9 we get k groups of 10 numbers showing the frequency of each number to be found at a "kth position" with possible additional shift from 0 to k-1.
If Pi is random we should find a distribution of all these groups of ten numbers near to the normal one.
It is quite enough to test p mod k for only prime numbers k, because a regularity in any other number will be reflected in the results for its prim factors.