In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along these lines known to date.
Deep Dive into Optimizing Properties of Balanced Words.
In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along these lines known to date.
Firstly, we recall that a finite or infinite 0-1 word w = w 1 w 2 . . . is called balanced if for every pair of finite subwords u, v such that |u| = |v|, we necessarily have ||u| 1 -|v| 1 | ≤ 1, where |u| 1 = #{ j : u j = 1} stands for the 1-length of u. An infinite balanced word which is not eventually periodic is called Sturmian.
There are several equivalent definitions of Sturmian sequences. Let w = w 1 w 2 . . . be an infinite 0-1 sequence and put p w (n) = #{w j . . . w j+n-1 : j ≥ 1} -the complexity function of w. Then w is Sturmian if and only if p w (n) = n + 1. The 1-ratio γ = lim n→∞ |w 1 . . . w n | 1 /n is well defined for any Sturmian sequence w; furthermore, any Sturmian sequence with the 1-ratio γ can be obtained by the formula
for some δ ∈ [0, 1). For more details see, e.g., [10]. This survey paper is concerned with some optimization problems from various areas of mathematics and physics, in which balanced words (and, in some cases, Sturmian sequences) turn out to be optimizing.
We will begin with optimization problems in mathematics. Our first example comes from the seminal paper [5].
Define vectors f 0 , f 1 , . . . , f m in Z m as follows:
Let J be a multimodular function on Z m and let J denote its lower convex envelope. If x is any infinite 0-1 sequence with the 1-ratio γ, then
where γ = (γ, . . . , γ). Moreover, if x = w given by (1) for some δ , then we have the equality.
The author then applies this result to the following queuing problem: consider a sequence of customers arriving at a fixed rate in such a way that the interarrival times are i.i.d. Poisson random variables with the same finite mean. A 0-1 input sequence x = (x 1 , x 2 , . . . ) determines what happens to the kth customer, namely, if x k = 1, the customer is admitted and if x k = 0, he is sent elsewhere. The mean service time is assumed to be fixed. The number in the queue after customer k arrives, is N k + x k . Then the expectation of max N k is a multimodular function.
Consequently, it follows from the above theorem that if a fraction γ of customers is sent to a server queue according to a splitting sequence x, then the long-term average is minimized when x k = ⌊(n + 1)γ⌋ -⌊nγ⌋, i.e., along Sturmian sequences.
Let w = w 1 . . . w m be a finite 0-1 word and put
where w (1) = w, w (2) , . . . , w (m) are the cyclic permutations of w. Let now W p,q denote the set of 0-1 words of length q with the 1-length p. As is well known (see, e.g., [10]), there are precisely q balanced words in W p,q , all of which are in the same orbit (= all cyclic permutations of a word), so if W p,q is defined to be the set of all orbits of words in W p,q , there is a unique balanced orbit in W p,q .
) Suppose 1 ≤ p < q are coprime integers. For w ∈ W p,q , the product B(w) is maximized precisely when w is balanced.
For instance, put p = 2, q = 5. Here there are only two possible orbits, namely, those of w = 10100 and v = 11000. We have
Let T : [0, 1) → [0, 1) and denote the space of all T -invariant measures by M (T ). We say that a measure µ is majorated by ν (notation: Here S γ is the following. Let ϕ : [0, 1) → [0, 1) be defined as follows:
i.e., the binary sum of the standard symbolic sequence associated with the rotation by γ with a starting point x. Then the Sturmian measure S γ is the push forward of the Lebesgue measure on [0, 1) under ϕ γ .
If γ is rational, then S γ sits on a finite set. For instance, the support of S 2/5 is the orbit of the binary sum of 00101 00101 00101 . . . under T , i.e., the set There are other papers in this area which produce Sturmian sequences in similar optimization problems. For instance, one may replace the class of convex functions with increasing functions and consider the β -transformation given by τ β x = β x mod 1 (with β > 1) and the β -shift -the subshift on the alphabet {0, 1, . . . , ⌈β ⌉ -1} which corresponds to the natural partition [0, 1)
It has been shown in [2] that the β -shift has a largest shift-invariant measure if and only if β is an algebraic integer of a special form. (In particular, if 1 < β < 2, then it has to be multinacci, i.e., the dominant root of x m = x m-1 + • • • + x + 1 for some m ≥ 2.) In this case the largest shift-invariant measure on the β -subshift is the unique one supported by the periodic shift-orbit generated by its lexicographically largest element. The supporting measure is always Sturmian.
Another direction in this line of research is concerned with imposing no extra conditions of the class of functions but instead considering a specific (usually, one-parameter) family of those. For example, let, as above, T be the doubling map, and let g θ (x) = cos 2π(x -θ ) or f θ (x) = 1 -4dist T (x, θ ), where dist T is the distance on the circle R/Z. In both cases maximizing measures are Sturmian -see [3,1] and references therein.
Thus, in questions concerning maximizing measures, the Sturmian measures seem to be a very robust class.
Consider a versio
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
