H. Eggleston, The fractional dimension of a set defined by decimal properties, Q. J. Math. 20 (1949) 31--36.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....concerning the size of these sets from the point of view of dimension theory is due to Besicovitch [19] For m = 2, he showed that if # (0, then [0, 1] lim sup # 1 (x, n) n # = # log # (1 #) log(1 . More detailed information was later obtained by Eggleston [39], who showed that dimH Fm (# 0 , #m 1 ) # k log m # k . 46) We note that it is easy to show and this does not require the above result that each set Fm (# 0 , #m 1 ) is dense in [0, 1] The identity in (46) can be established by applying Theorem 11 when m = 2, Theorem ....

H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31--36.

....sets from the point of view of dimension theory is due to Besicovitch [4] For m = 2, he showed that if # ) then [0, 1] lim sup # 1 (x, n) n # #) log(1 where dimH Z denotes the Hausdor# dimension of the set Z. More detailed information was later obtained by Eggleston [8], who showed that . 3) An immediate consequence is that if # i (0, 1) for some i, then the set Fm (# 0 , #m 1 ) is nonempty (and thus dense in [0, 1] with uncountable many points and even positive Hausdor# dimension. The work of Eggleston was further generalized by Billingsley ....

....that for each choice of functions # i one can explicitly exhibit a measure sitting on the level set K # , in the sense that (32) holds. In the case of 1 locally constant functions these measures are always Bernoulli measures. A first consequence of Theorem 11 is the classical result of Eggleston [8] described in the introduction. Corollary 12. For every (# 0 , #m 1 ) Lm , k=0 # k log # k . Proof. Setting # k and # k as in (20) for k = 0, m 1, we obtain K # = Fm (# 0 , #m 1 ) The statement follows immediately from Theorem 11. Theorem 11 allows us to ....

H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31--36.

.... #m 1 = 1 with # i # [0, 1] for each i. One can show that each of these sets is nonempty and hence is dense in [0, 1] note that the limits # k (x) only depend on the tail of the representation) In fact it is straightforward to construct explicitly a point in Fm (# 0 , #m 1 ) In [5] Eggleston computed the Hausdor# dimension dimH Fm (# 0 , #m 1 ) # m 1 k=0 # k log # k log m . 3) It is easy to see that this result is related to multifractal analysis. Observe first that the action of the shift map on the set of sequences in 0, m 1 can be identified ....

H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31--36.

....for the probabilistic sources (C. Shannon [8] Csiszar and Korner [2] etc. We analyze the performance of the optimal code and express it in terms of Hausdorff dimension. It is interesting to note that a close relationship between information theory and Hausdorff dimension was discovered [3] as soon as C. E. Shannon [8] published his classical paper [8] Let A = fa 1 ; a 2 ; Delta Delta Delta ; a r g and B = fb 1 ; b 2 ; Delta Delta Delta ; b s g be source and channel alphabets, respectively. We define a This work was done while Boris Ryabko was staying in Osaka, Japan in 2000 ....

H. G. Eggleston, "The fractional dimension of a set defined by decimal properties", Quart. J. Math. Oxford Ser., vol. 20, No. 1, pp. 31-36 (1949).

....that the formula dimension = entropy Lyapunov exponent has been established in many settings, but usually in the cases when there is no overlap, see, e.g. PARABOLIC ITERATED FUNCTION SYSTEMS 3 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Figure 1. The iterated function system f x 1=4 x 5=4 ; x x 1 g [Eg, KP, Mn, Y, Ma, MU]. Stationary measures for 2 Theta 2 random matrices (of which ff is an example) and their dimension properties, have been investigated by Ledrappier [L] see also [BL] Before [Ly] Pincus [Pi] studied Bernoulli random matrices and their stationary measures using the IFS approach; he found ....

H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford, Ser. 20 (1949), 31--36.

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H. Eggleston, The fractional dimension of a set defined by decimal properties, Q. J. Math. 20 (1949) 31--36.

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H.G. Eggleston. The fractional dimension of a set defined by decimal properties. Quarterly Journal of Mathematics, Oxford Series 20:31--36, 1949.

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H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford, Ser.20 (1949), 31--36.

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