Chiffres non nuls dans le développement en base entière des nombres algébriques irrationnels

On the beta-expansions of 1 and algebraic numbers for a Salem number beta

“…Changing C 2 (η) and C 3 (η) by suitable positive constants C 2 (b, η) and C 3 (b, η), respectively, we can prove (1.5) for any integral base b in the same way as in the case of b = 2. Moreover, modifying the proof of (1.5), Adamczewski and Faverjon [4], and Bugeaud [11] independently gave effective versions of the lower bounds for general integer b ≥ 2. Here we introduce the results by Bugeaud as follows.…”

On the binary digits of $\sqrt{2}$

“…where nz(N ) is the number of non-zero digits in the first N binary digits after the decimal point of √ 2. However, we cannot yet prove a result of the strength of (1). Indeed, the strongest result we have available is nz(N ) ≥ N 1/2 (1 + o(1)), due to Bailey, Borwein, Crandall, and Pomerance [2].…”

On the size of a restricted sumset with application to the binary expansion of √d

“…In [3], it was shown that the binary expansion of an algebraic number β of degree deg β ≥ 2 has at least c(β)N 1/ deg β units among the first N digits. Various generalizations of this result have been given in [1], [7], [8], [9], [11] (see also [5,Theorem 8.5]). In particular, in [1] and [5] it was shown that the constant c(β) is effectively computable.…”

“…Various generalizations of this result have been given in [1], [7], [8], [9], [11] (see also [5,Theorem 8.5]). In particular, in [1] and [5] it was shown that the constant c(β) is effectively computable. From the proof one can see that the constant N 0 (d) of Theorem 3 is effectively computable.…”