The following problem has been open since 1985: Does there exist an infinite word w over a finite set of non-negative integers such that w does not contain any two consecutive blocks with the same length and the same sum? This problem was considered independently by Brown and Freedman (in 1987), Pirillo and Varricchio (in 1994), and Halbeisen and Hungerbuher (in 2000). We show that the answer is “no” for all 4-element sets {a, b, c, d} where a < b < c < d are real numbers satisfying the Sidon equation a + d = b + c. For any finite subset T of R, we define g(T) to be the maximum length of a word over T which does not contain any two consecutive blocks with the same length and the same sum. (We allow g(T) = 1.) In general, very little is known about g. Here we find the exact values of g(T) for all 4-element sets of real numbers T = {a, b, c, d}, a + d = b + c. We also show that g(T) 50 for all 4-element sets of real numbers, with equality if and only if T is an arithmetic progression.
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