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Let D n be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size µ D n (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of D n of cardinality r and s respectively. It is shown by construction that µ D n (r, s) is bounded above by the known… (More)

Eliahou, S., The 3x+ 1 problem: new lower bounds on nontrivial cycle lengths, Discrete Mathematics 118 (1993) 45556. Let 7': N-+ N be the function defined by T(n) = n/2 if n is even, T(n) = (3n + 1)/2 if n is odd. We show, among other things, that any nontrivial cyclic orbit under iteration of T must contain at least 17 087 915 elements.

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in Z/pZ, the cardinality of the sumset A + B = {a + b | a ∈ A, b ∈ B} is bounded below by min(r + s − 1, p); moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers r, s ≤… (More)

Steinhaus graphs on n vertices are certain simple graphs in bijec-tive correspondence with binary {0,1}-sequences of length n − 1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110...110 of any length n − 1 = 3m. By an exhaustive search the conjecture was… (More)