Fraenkel's Conjecture

The set B(x,y)={ Floor[ n x + y ] : n \in Z } is approximately an arithmetic progression. A. Fraenkel considered the problem of partitioning the integers into finitely many B(x,y) sets, particularly with distinct values of x. There are many ways to do this with exactly two parts, and he found for each m>2 a single way to partition Z into exactly m B(x,y) sets with distinct x?s. He conjectured that his examples were the only way to accomplish a partitioning of Z into B(x,y) sets with distinct x. Two special cases are noteworthy. If one assumes that all of the x?s in a partition are integers, then basic generating functionology shows that two of the x?s must be equal. If one assumes that at least one of the x?s is irrational, then basic uniform distribution results imply that two of the x?s must be equal.

This problem is notable because (like the Lonely Runner Conjecture), the integer and irrational cases are both easier than the "middle" case: rationals. I will discuss these results and recent progress on generalizing the conjecture. This is joint work with Ron Graham.

Kevin O'Bryant