Irrational algebraic numbers are conjectured to be normal in all basis. This implies in particular that 1 should appear with a frequency 1/2 in their binary expansion but even this is unknown for “natural” numbers. One way to approach this problem is to prove lower bounds for the bit counting function , the number of 1 in the first n digits of the binary expansion.
For or , it is known that for large (by Bundschuh (1971) and Davis (1978)).
(Thanks to J.-P. Allouche)