Paper Folding, Digit Patterns and Groups of Arithmetic Fractals

Digit patterns and transcendental numbers

“…We apply the Loxton and van der Poorten theorem to certain groups of sequences that we investigated in [13]. These groups are defined as follows.…”

“…The moduli M for which (4) holds are then exactly the multiples of M(a). In [13] it is shown that T k (G) is a group under componentwise multiplication (addition, if G is additive), and that the sequences for which M(a) divides a power of k form a subgroup A k (G).…”

“…We remark that our methods here, as in [13], are elementary. The Loxton and van der Poorten theorem is the only non-elementary component of our argument.…”

“…Then M{{Y n }) = 1, Y o is the identity in G = {C.f, and {Y n } has least period p q . Now let (p { \ G -» (Q be the projection homomorphism of G which takes a vector to its rth coordinate, and define the sequence a ; by Since <p t : G -> (C), a, e ^((C)) by results of [13] (equations (34) and (34')). It is easily checked that M(a ( ) = 1 and that a, is periodic with By results of [13] (see Theorem 7), M(a ( ) = 1 implies that a, is a product of pattern sequences a , where j runs over a set S of digits base k: REMARK.…”

“…Now let (p { \ G -» (Q be the projection homomorphism of G which takes a vector to its rth coordinate, and define the sequence a ; by Since <p t : G -> (C), a, e ^((C)) by results of [13] (equations (34) and (34')). It is easily checked that M(a ( ) = 1 and that a, is periodic with By results of [13] (see Theorem 7), M(a ( ) = 1 implies that a, is a product of pattern sequences a , where j runs over a set S of digits base k: REMARK. Exactly the same proof shows that any ultimately periodic sequence in A fc (Z) is purely periodic with period equal to M(a), a power of k, and that a periodic sequence a in A fc (Z r ) has a least period of the form bM(a), where 6|(r,fc-1).…”