Warren Dicks and J. Porti,
On the Hausdorff dimension of the Gieseking fractal.
Topology Appl., 126 (2002), 169-186.

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                                                        Addenda

                                                    May 1, 2005.

In this article, we studied the Hausdorff dimension  d  of the Gieseking fractal.  We can also say
that  d  is the Hausdorff dimension of the Gieseking-fractal arcs that tessellate the Euclidean plane
like this:

There is a computer experiment which currently suggets that  d   is about 1.29710... .
Let  S  denote the free semigroup on the two-letter alphabet {g,  f }.  Let  S  act on  N⁴  by
         g ( a, b, c, d )  =  ( a, b + d, a + 2b + c, d ),
         f ( a, b, c, d )  =  ( a + b, c, b, a + c + d ).
We define the weight of an element  ( a, b, c, d )  of  N⁴  to be  b + c + d.  
For each  n  in  N,  let   b_n  denote the number of words  w  in  S  such that the weight of
w ( 0, 0, 1, 1 )  is at most  2 cosh(n).  If  n  is positive,  let  a_n = b_n - b_n-1.   Recall that  d  lies
between the liminf and the limsup of   ( log  a_n/a_n-1  |  n = 2, 3, ... ).   We do not know whether or
not this sequence converges.   Here are the first 24 terms.

n	bn	an	log an/an-1	Programmer	Year
0	1
1	4	3		W. Dicks	2000
2	11	7	0.8472978	W. Dicks	2000
3	38	27	1.3499267	W. Dicks	2000
4	153	115	1.4490952	W. Dicks	2000
5	526	373	1.1766462	W. Dicks	2000
6	1946	1420	1.3368337	W. Dicks	2000
7	7111	5165	1.2912482	W. Dicks	2000
8	25882	18771	1.2904080	W. Dicks	2000
9	95156	69274	1.3057565	W. Dicks	2000
10	348335	253179	1.2960270	A. Dicks	2000
11	1271898	923563	1.2941422	A. Dicks	2000
12	4658192	3386294	1.2992523	A. Dicks	2000
13	17040856	12382664	1.2965613	A. Dicks	2000
14	62350742	45309886	1.2972278	A. Dicks	2000
15	228147404	165796662	1.2972368	A. Dicks	2000
16	834711907	606564503	1.2970489	J.- C. Artés	2000
17	3053925244	2219213337	1.2970969	P. Taylor	2000
18	11172970200	8119044956	1.2970597	P. Taylor	2000
19	40878223775	29705253575	1.2971113	P. Taylor	2000
20	149563132885	108684909110	1.2971290	P. Taylor	2000
21	547206912858	397643779973	1.2971036	J. Amoros	2003
22	2002063012565	1454856099707	1.2971056	J. Amoros	2003
23	7324919842341	5322856829776	1.2971031	J. Amoros	2003
24	26799555593731	19474635751390	1.2971027	J. Amoros and  J. Vindel	2004
25	98051175064382	71251619470651	1.2971046	J. Vindel	2004

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By left conjugating by   h( a, b, c, d )  =  ( a, b, d, b + c + d )  we can rewrite the above procedure as
follows.

 Let  S  denote the free semigroup on the two-letter alphabet {g,  f }.  Let  S  act on  Z⁴  by
         g ( a, b, c, d )  =  ( a, b + c,  c,  a + 2b + c + d ) 
                                =  ( a, b + c,  c,  (b + c) + d + (a + b)) ,
         f ( a, b, c, d )  =   ( a + b, - b - c + d,  a - b + d,  a - b - c + 2d )
                                =   ( a + b, - (b + c) +  d,  a - b + d, - c +  d + (a - b+ d )).
For each  n  in  N,  let   b_n  denote the number of words  w  in  S  such that the fourth coordinate of
w ( 0, 0, 1, 2 )  is at most  2 cosh(n).  If  n  is positive,  let  a_n = b_n - b_n-1.   Then  d  lies
between the liminf and the limsup of   ( log  a_n/a_n-1  |  n = 2, 3, ... ).   We do not know whether or
not this sequence converges.   The first 24 terms are as above.

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