Some Sets of GCF ϵ Expansions Whose Parameter ϵ Fetch the Marginal Value

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1 Journal of Mathematical Reearch with Application May, 205, Vol 35, No 3, pp DOI:03770/jin: Some Set of GCF ϵ Expanion Whoe Parameter ϵ Fetch the Marginal Value Liang TANG, Peijuan ZHOU, Ting ZHONG Department of Mathematic, Jihou Univerity, Hunan , P R China Abtract Let ϵ : N R be a parameter function atifying the condition ϵ) + + > 0 and let T ϵ : 0, ] 0, ] be a tranformation defined by T ϵ x) = + + )x + ϵx for x +, ] Under the algorithm T ϵ, every x 0, ] i attached an expanion, called generalized continued fraction GCF ϵ ) expanion with parameter by Schweiger Define the equence { n x)} n of the partial quotient of x by x) = /x and n x) = Tϵ n x)) for every n 2 Under the retriction < ϵ) <, define the et of non-recurring GCF ϵ expanion a F ϵ = {x 0, ] : n+x) > nx) for infinitely many n} It ha been proved by Schweiger that F ϵ ha Lebegue meaure 0 In the preent paper, we trengthen thi reult by howing that { dimh F ϵ, when ϵ) = + ρ for a contant 0 < ρ < ; 2 dim +2 H F ϵ, when ϵ) = + for any where dim H denote the Haudorff dimenion Keyword GCF ϵ expanion; metric propertie; Haudorff dimenion MR200) Subject Claification K55; 28A80 Introduction In 2003, Schweiger [] introduced a new cla of continued fraction with parameter, called generalized continued fraction GCF ϵ ), which are induced by the tranformation T ϵ : 0, ] 0, ] T ϵ x) := where the parameter ϵ : N R atifie + + )x + ϵ ϵx when x +, ] =: B) ) ϵ) + + > 0, for all 2) Received July 24, 204; Accepted December 22, 204 Supported by the National Natural Science Foundation of China Grant No 36025) * Correponding author addre: tl3022@26com Liang TANG); peijuanzhou@26com Peijuan ZHOU); zhongting 2005@26 com Ting ZHONG)

2 Some et of GCF ϵ expanion whoe parameter ϵ fetch the marginal value 257 For any x 0, ], it partial quotient { n } n in the GCF ϵ expanion are defined a = x) :=, and n = n x) := T n ϵ x) x ) By the algorithm ), it follow [] that x = A n + B n T n ϵ x) C n + D n T n ϵ x) for all n, where the number A n, B n, C n, D n are given by the recurive relation ) ) C n D n C n D n n + n ϵ n ) = A n B n C n B n + ϵ n ) ) ) C 0 D 0 0 with = A 0 B 0 0 ) n, 3) A well own example of the generalized continued fraction i in the cae that the parameter function ϵ 0 In thi cae, the algorithm ) become T x) = + + )x when x +, ] Then every x 0, ] can be expanded into a erie with the form x = x) ) 2 x) + ) n x) + ) + Actually thi i the Engel erie expanion which wa well tudied in the literature, ee Erdö, Rényi & Szüz [2], Rényi [3], Galambo [4] and Liu, Wu [5], etc Schweiger [] tudied the arithmetical a well a the ergodic propertie of GCF ϵ map At the ame time, he howed that with different choice of the parameter function ϵ, the tochatic propertie of the partial quotient differ greatly Concerning the propertie of the partial quotient, by the condition hared by the parameter ϵ) ee 2)), it i clear that n+ x) n x) for all n, ie, the partial quotient equence of x i non-decreaing We invetigated the metrical propertie of { n } n further in [8] and proved that when < ϵ), for almot all x 0, ], log n x) lim =, n and when ϵ) =, thi equality i no longer true It wa alo hown [7] that the partial quotient in the GCF ϵ expanion hare a 0- law and the central limit theorem under the retriction of < ϵ) Thee reult howed that when < ϵ), the metric propertie of GCF ϵ and Engel erie expanion are very imilar However, in thi paper we will ee that the ituation change radically when < ϵ) < ρ for a contant 0 < ρ < Thi i becaue in thi cae, T ϵ ha two fixed point ϵ and in every interval B) := +, ] So all B) can be divided into two ubinterval a: [ B ) =: +, ] and B + ) =: ϵ) ϵ), ]

3 258 Liang TANG, Peijuan ZHOU and Ting ZHONG uch that T B + ) = B + ) Therefore if, 2,, n, + ) i an admiible bloc, then n < And it i eay to ee that, the et defined by F ϵ = B, 2,, n ) 4) n= n i a complementary et of the ultimately recurring GCF ϵ expanion That i F ϵ := {x 0, ] : n+ x) > n x) for infinitely many n} We define the cylinder et a follow For any non-decreaing integer vector,, n ), define the n-th order cylinder a follow B,, n ) = {x 0, ] : j x) = j, j n} an nth order cylinder, which i the et of point whoe partial quotient begin with,, n ) Then the following reult have been obtained in ection 3 of []: B, 2,, n ) = B nc n A n D n C n n C n + D n ) ; 5) B, 2,, n ) B n C n A n D n = C n ϵ n )C n + D n ) ; 6) λ ) ) F ϵ = λ B, 2,, n ) = 0, 7) n= < < n where < ϵ) < + ρ for a contant 0 < ρ < In thi paper, we trengthen the reult 7) by howing that Theorem Let F ϵ be the et defined above Then { dim H F ϵ 2, when ϵ) = + ρ for a contant 0 < ρ < ; +2 dim H F ϵ, when ϵ) = + for any where dim H denote the Haudorff dimenion 2 Preliminary later ue In thi ection, we preent ome imple fact about the generalized continued fraction for The firt lemma concern the relationhip between A n, B n, C n, D n which are recurively defined by 3) Lemma 2 [,8]) For all n, i) C n = n + )C n + D n > 0; ii) D n = n ϵ n )C n + + ϵ n ))D n, and D n 0 when ϵ 0; D n < 0 when ϵ < 0; iii) n C n + D n = n C n + D n ) n + + ϵ n )); iv) B n C n A n D n = B N C N A N D N ) n i=n+ i + + ϵ i )) > 0, 0 N < n The following lemma are epecially aimed for ϵ) = +

4 Some et of GCF ϵ expanion whoe parameter ϵ fetch the marginal value 259 Lemma 22 If ϵ) = +, then when n 2, n C n + D n = nc n + D n n > 0; ϵ n )C n + D n C n 2 Proof By Lemma 2 iii) and the condition ϵ) = +, noticing that n n, we have n C n + D n = nc n + D n n Thi alo give that Uing Lemma 2 i) and 8), we get n C n + D n n C + D n n 2 > 0 D n n C n 8) C n n + )C n n C n n + n )C n C n C = + 2 Thu C n 2 i proved Uing 8) again, we can find that when n 2, ϵ n )C n + D n n + )C n n C n = )C n 2 C n The next lemma will be ued for etimating the lower bound of dim H F ϵ Lemma 23 [,8]) Let ϵ) = + Then when n 2, C n + D n 0; n C n + D n ϵ n )C n + D n C n n n Proof By Lemma 2 i) ii), we have C n + D n = n + )C n + D n + n n + )C n + n + )D n = C n + D n + n + ) n C n + D n ) Then by Lemma 2 i), we have Thu n C n + D n n C n + D n ) 0 C n = n C n + C n + D n ) n C n n n 9) By the condition ϵ) = +, we have, n < ϵ n ) = n + n C n + D n ϵ n )C n + D n = n C n + D n ) + )C n 0) Then uing the firt equality of Lemma 22, we get ϵ n )C n + D n = nc n + D n n So the econd reult follow from 0), ) and 9) + )C n = C n C n C n )

5 260 Liang TANG, Peijuan ZHOU and Ting ZHONG Now we focu on the propertie of the point et F ϵ with ϵ) = + for any n From now on until the end of thi paper, we fix a point x F ϵ and let n = n x) be the nth partial quotient of x The number A n, B n, C n, D n are recurively defined by 3) for x 3 The Haudorff dimenion of E ϵ α) The proof of Theorem i divided into two part: one for upper bound, the other for lower bound 3 Upper bound Fix δ > 0 Since n = that for all j M, ) +δ = n= n= j n n +δ ) +δ converge, there exit M large enough o 2) From 4), we can ee that n B, 2,, n ) i a natural covering of F ϵ for any n Then the +δ -dimenional Haudorff meaure of F ϵ can be etimated a H +δ Fϵ ) lim inf i+ i i n B, 2,, n ) Under the condition ϵ) = +, by Lemma 2 iv), we have B n C n A n D n = +δ 2 n ) 3) On the other hand, by Lemma 22, we have C n ϵ n )C n + D n ) 2 Then uing 6), we get B, 2,, B n C n A n D n n ) = C n ϵ n )C n + D n ) 2 2 n ) 2 2 N ) N+ N+2 Thu by 2), we have H +δ Fϵ ) lim inf lim inf lim inf i+ i i N i+ i i N i+ i i n B, 2,, n ) ) +δ 2 N ) N+ N ) +δ 2 N ) < +δ N+ ) +δ n n which give that dim H E ϵ α) +δ Since thi i true for all δ > 0, we get dim H E ϵ α) for ϵ) = + and any n ) +δ

6 Some et of GCF ϵ expanion whoe parameter ϵ fetch the marginal value Lower bound In order to etimate the lower bound, we recall the claical dimenional reult concerning a pecially defined Cantor et Lemma 3 [6]) Let I = E 0 E E 2 be a decreaing equence of et, with each E n, a union of a finite number of dijoint cloed interval If each interval of E n contain at leat m n interval of E n n =, 2, ) which are eparated by gap of at leat η n, where 0 < η n+ < η n for each n Then the lower bound of the Haudorff dimenion of E can be given by the following inequality: dim H n E n ) lim inf logm m 2 m n ) logm n η n ) Now for each n, let E = {x 0, ] : 2 n < n x) < 2 n+, n } Clearly, if x E, then n x) > n x) for all n Thi implie that E F ϵ For each n, let E n be the collection of cylinder E n = {B n,, n ) : 2 i < i x) < 2 i+, i n} 4) Then E = n= E n, and E fulfill the contruction of the Cantor et in Lemma 3 Now we pecify the integer {m n, n } and the real number {η n, n } E n, and Due to the definition of E n, each interval of E n contain m n = 2 n 2 n interval of m m 2 m n = n 2) = 2 n 2)n ) 2 ; 5) and any two of interval in E n are eparated by at leat one interval B,, n, + n ) By 5) and 6), we have B,, n, + n ) = B,, n, n ) B,, n, n ) = B nc n A n D n C n n C n + D n ) B n C n A n D n C n ϵ n )C n + D n ) = B nc n A n D n ) ϵ n ) n ) n C n + D n ) ϵ n )C n + D n ) By Lemma 23 and the equality 3), the above equality give that B,, n, + n ) 2 n ) +2 In view of 4), the partial quotient n atify that 2 n < n x) < 2 n+ for all n Therefore, B,, n, + n ) A a reult of 5) and 6), we get lim inf n+) ) = +2 2 nn+)+2) 2 log 2 m m n ) log 2 m n η n ) = + 2 Combining thi with Lemma 3, we get when ϵ) = + for any, dim H F ϵ dim H E + 2 =: η 6)

7 262 Liang TANG, Peijuan ZHOU and Ting ZHONG Uing the ame method of proof, we can get dim H F ϵ 2 contant 0 < ρ < when ϵ) = + ρ for a Acowledgement We than the referee for their time and comment Reference [] F SCHWEIGER Continued fraction with increaing digit Öterreich Aad Wi Math-Natur Kl Sitzungber II, 2003, 22: [2] P ERDÖS, A RÉNYI, P SZÜSZ On Engel and Sylveter erie Ann Univ Sci Budapet Eötvö Sect Math, 958, : 7 32 [3] A RÉNYI A new approach to the theory of Engel erie Ann Univ Sci Budapet Eötvö Sect Math, 962, 5: [4] J GALAMBOS Reprentation of Real Number by Infinite Serie Lecture Note in Math Vol502, Berlin, Springer, 976 [5] Yanyan LIU, Jun WU Some exceptional et in Engel expanion Nonlinearity, 2003, 62): [6] K J FALCONER Fractal Geometry Mathematical Foundation and Application Wiley, 990 [7] Luming SHEN, Yuyuan ZHOU Some metric propertie on the GCF fraction expanion J Number Theory, 200, 30): 9 [8] Ting ZHONG Metrical propertie for a cla of continued fraction with increaing digit J Number Theory, 2008, 286):