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Toward Normality of Algebraic Numbers?

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 $B(n,x)$, the number of 1 in the first n digits of the binary expansion.

For $\pi$ or $e$, it is known that $B(x,n)\geq C \log n$ for large $n$ (by Bundschuh (1971) and Davis (1978)).

Building on Bailey, Borwein, Crandall and Pomerance, Rivoal has recently proved that $B(x,n)\geq C n^{1/d}$ for all large $n$ when $x$ is an algebraic number of degree $d$.

(Thanks to J.-P. Allouche)