Fixed-Point Problems in Discrete Tomography: Case of Square Windows

A kind of fixed-point problem in the area of discrete tomography is proposed and investigated. Our chief concern in this paper is the case of square windows in the plane. Dealing with the arrays which are bounded, of polynomial growth, and finite-ring-valued, one comes across several interesting phenomena of combinatorial and arithmetic nature. keywords: discrete tomography; fixed point; square window; balanced array 0 Introduction In the articles [2], [3], we show that various problems in the area of discrete tomography, for example as in [4], can be understood through the theory of distributions in a unified way. The main purpose of the present paper is to investigate several combinatorial phenomena which show up when we try to apply the general theory developed there to concrete problems. In particular we focus on the case of square windows and investigate what kind of arrays arise as solutions to tomographic problems. The shapes of solutions depend heavily on the conditions which we impose on the arrays. Accordingly we divide our presentation into several sections which deal with arrays that are bounded, of polynomial growth, Fp-valued, and balanced. The plan of the paper is as follows. In Section one, after we fix some notation and recall the main results of [2] and [3], we formulate the problem which is of our major concern throughout the paper. Section two is devoted to the study of the characteristic polynomials attached to the square and related windows. As a result we determine the structure of the solution spaces of arrays which are bounded. In Section three we consider our problem for the arrays of polynomial growth employing the tools developed in [3]. Section four deals with the case of Fp-valued arrays, and Section five concerns with the case of Zn2 -valued balanced arrays. Here we come across to a problem which resembles to the ”Sudoku” game. In this final section we propose several open problems to the reader. 1 Problem Setting. For any commutative ring R, let A(R) = (R)Z 2 denote the set of R-valued functions on Z. We write elements of A(R) as a = (ai) with i = (i1, i2) ∈ Z, and call them R-valued arrays on Z. For any non-empty finite subset W of Z and for any R-valued array a ∈ A(R), let dW (a) = ∑ i∈W ai ∈ R, and call it the degree of a with respect to W . For any k ∈ Z, let W+k = {i+k; i ∈ W} denote the translation of W by k. By getting dW+k(a)(k ∈ Z) together, we obtain a new R-valued array (dW+k(a))k∈Z2 . We denote this new array by ∆W (a) so that the set of R-valued arrays are equipped with a self-map ∆W : A(R) → A(R), which is easily seen to be an R-endomorphism through the natural R-module structure on A(R). The main purpose of this paper is to investigate the set of fixed points of ∆W for various rings and finite subsets W ⊂ Z. Accordingly we introduce the following notation: Definition 1.1. For any non-empty finite subset W of Z, let FixW (R) denote the set of fixed points of ∆W : A(R) → A(R), namely FixW (R) = {a ∈ A(R);∆W (a) = a}. By the very definition of ∆W , we have the equality FixW (R) = FixW+k(R) for any k ∈ Z. Therefore we assume throughout the paper that W contains the origin O = (0, 0) ∈ Z, and let W ∗ = W \{O}. The following simple observation provides us with a connection between our problem and the results obtained in [2] and [3]: Proposition 1.1. For any non-empty finite subset W of Z, we have FixW (R) = {a ∈ A(R);∆W∗(a) = 0}, (1.1) where 0 denotes the all-zero array. We will denote the set on the right hand side of (1,1) by AW∗ . Here we recall one of the main results in [2]. For any window W , we put mW (x, y) = ∑ (i1,i2)∈W x1y2 ∈ Z[x±1, y±1], and call it the characteristic (Laurent) polynomial of W . For any Laurent polynomial f , we define f∗ by the following rule: f∗(x, y) = f(1/x, 1/y). 2 Furthermore we define a subset AW (C) of AW (C) by AW (C) = {a ∈ AW (C); there exists a constant C such that |ai| < C(i ∈ Z)}. Then we have the following equality [2, Theorem 3.2] dimC AW (C) = #(VT2(mW )), (1.2) where T = {z ∈ C; |z| = 1} and VT2(mW ) denotes the zero locus of mW on T. 2 Square window For any integer n ≥ 2, let S(n) denote the subset of Z defined as S(n) = {(i1, i2) ∈ Z; 0 ≤ i1, i2 ≤ n− 1}, and call it the square window of width n. Its characteristic polynomial mS(n) is given by mS(n)(x, y) = (1 + x+ x + · · ·+ xn−1)(1 + y + y + · · ·+ yn−1), hence that of S(n)∗ is given by mS(n)∗(x, y) = (1 + x+ x + · · ·+ xn−1)(1 + y + y + · · ·+ yn−1)− 1. For any positive integer N , we denote by μN the set of N -th roots of unity in C, and let μN = μN \ {1}. Proposition 2.1. For any c ∈ μn−1 let S1(c) = {(x, y) ∈ (μn−1);xy = c}, S2(c) = {(x, y) ∈ (μ(n−1)(n+1)) ;xy = c, x = c}. Then we have VT2(mS(n)∗) = ∪ c∈μn−1 (S1(c) ∪ S2(c)) . Proof. Since mS(n)∗ is defined over R, if (x0, y0) ∈ VT2(mS(n)∗) then its complex conjugate (x0, y0) is also in VT2(mS(n)∗). Furthermore since we have z̄ = 1/z for any z ∈ T, we see that VT2(mS(n)∗) = VT2(mS(n)∗) ∩ VT2(mS(n)∗). Therefore we are to solve the following simultaneous equations: (1 + x+ x + · · ·+ xn−1)(1 + y + y + · · ·+ yn−1)− 1 = 0, (2.1) (1 + x−1 + x−2 + · · ·+ x−(n−1))(1 + y−1 + y−2 + · · ·+ y−(n−1))− 1 = 0. (2.2) By multiplying (2.2) by xn−1yn−1 and subtracting it from (2.1), we obtain the equality xn−1yn−1 = 1. 3 Hence there exists an element c ∈ μn−1 such that y = c/x. First we deal with the case when c 6= 1. By inserting y = c/x into the equation (2.1) and expanding it, we can compute the left hand side as follows: (1 + x+ x + · · ·+ xn−1) +c(x−1 + 1 + x+ · · ·+ xn−2) +c2(x−2 + x−1 + 1 + · · ·+ xn−3) · · · · · · · · · +cn−1(x−(n−1) + x−(n−2) + x−(n−3) + · · ·+ 1)− 1 = xn−1 + 1 ∑ k=0 ckxn−2 + · · ·+ ( n−2 ∑