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)).

Building on Bailey, Borwein, Crandall and Pomerance, Rivoal has recently proved that for all large when is an algebraic number of degree .

(Thanks to J.-P. Allouche)

## Leave a Reply