Equation-regular sets and the Fox-Kleitman conjecture
Abstract Given k ≥ 1 , the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2 k -variable linear Diophantine equation ∑ i = 1 k ( x i − y i ) = b is ( 2 k − 1 ) -regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1 , this equation is not 2 k -regular. While the conjecture has recently been settled for all k ≥ 2 , here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1 . In particular, this independently confirms the conjecture for k = 3 . We also briefly discuss the case k = 4 .
We consider families of operators satisfying a general class of relations, whose solutions can be described in terms of orbits of some dynamical system acting on the spectrum of a commuting sub-family. In the first section we introduce a class of relations and show, how the representations of such relations are related to orbits of the corresponding dynamical system. Also, we discuss the problem of accurate sense of the relation for unbounded operators. In Section 2, we study the class of *-algebras allowing Wick ordering whose representations can be studied by using methods of Section 1. We classify such Wick *-algebras, and discuss their representations.