Let \a = (a_1,a_2,...) \in {0,1,2}^\N be a 3m-periodic sequence (i.e., a_{j+3m}=a_j for all j\in\N).
ASSUME that # {1 \le r \le 3m : a_r = i}=m for i=0,1, and 2
(i.e. the 0's, 1's, and 2's appear the same number of times in each period).
For i=1,2 define the function U^i:\N \to \N by
U^i(x):= # of i's in the sequence \a prior to x appearances of 0
(i.e. let T(x) = min{t: \sum_{r=1}^t 1_{a_r=0} =x},
then U^i(x)=\# {1\le r \le T(x): a_r=i} ).
Let U^i_j, i=1,2 and j=1,...,m, be the function
U_i for the j-shifted sequence \a_j := (a_j,a_{j+1},...).
Is there always some 1
Tal Orenshtein
A counter example with m=5 was found, see:
http://mathoverflow.net/questions/233608/combinatorics-on-cyclic-sequences/234547#234547
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