No abstract

It is known that if a subset of R \mathbb {R} has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each λ \lambda in [ 0 , 1 ) [0,1) , we construct a subset of R \mathbb {R} that intersects every interval of unit length in a set of measure at least λ \lambda , but that does not contain any infinite arithmetic progression.

The literature contains two different classifications of solvable Lie algebras of dimensions up to and including 4. This paper is devoted to comparing the two classifications and translating each into the other. In particular, we exhibit an isomorphism between each solvable Lie algebra of one classification and the corresponding algebra of the second. The first classification is provided by de Graaf, and the second classification is from a recent book by Šnobl and Winternitz.