  We prove a quantitative version of the Polynomial Szemeredi Theorem for
difference sets. This result is achieved by first establishing a higher
dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of
the Polynomial Szemeredi Theorem asserting that the difference set of any
subset of the integers of positive upper density necessarily contains a perfect
square) and then applying a simple lifting argument.

Classical Analysis and ODEs;Number Theory