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Complexité des suites de Rudin-Shapiro généralisées

J.-P. AlloucheJ. O. Shallit — 1993

Journal de théorie des nombres de Bordeaux

La complexité d’une suite infinie est définie comme la fonction qui compte le nombre de facteurs de longueur k dans cette suite. Nous prouvons ici que la complexité des suites de Rudin-Shapiro généralisées (qui comptent les occurrences de certains facteurs dans les développements binaires d’entiers) est ultimement affine.

New bounds on the length of finite pierce and Engel series

P. ErdösJ. O. Shallit — 1991

Journal de théorie des nombres de Bordeaux

Every real number x , 0 < x 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually...

Linear fractional transformations of continued fractions with bounded partial quotients

J. C. LagariasJ. O. Shallit — 1997

Journal de théorie des nombres de Bordeaux

Let θ be a real number with continued fraction expansion θ = a 0 , a 1 , a 2 , , and let M = a b c d be a matrix with integer entries and nonzero determinant. If θ has bounded partial quotients, then a θ + b c θ + d = a 0 * , a 1 * , a 2 * , also has bounded partial quotients. More precisely, if a j K for all sufficiently large j , then a j * | det ( M ) | ( K + 2 ) for all sufficiently large j . We also give a weaker bound valid for all a j * with j 1 . The proofs use the homogeneous Diophantine approximation constant L θ = lim sup q q q θ - 1 . We show that 1 det ( M ) L ( θ ) L a θ + b c θ + d det ( M ) L ( θ ) .

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