Certain sequences of Wythoffian matrices, and maximal geometric progressions therein
Van der Waerden's Theorem states that if the positive integers are partitioned in two classes then at least one of the classes must contain arbitrary long arithmetic progressions. Several authors extended this theorem to establish the existence of long arithmetic progressions in different classes of ``not too thin'' sets of positive integers. Since the (additive) group \(\mathbb{Z}\) of integers can be represented (multipicatively) through a subgroup of \(\text{Gl}_ 2(\mathbb{Z})\) one further extension of van der Waerden type problems is the study of geometric progressions in subsets of \(\text{Gl}_ 2(\mathbb{Z})\). For a particular subset arising naturally in the study of winning pairs for ``Wythoff's Nim'' the author determines the distribution of maximal geometric progressions.
scientific article; zbMATH DE number 679881
scientific article