Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions

An odd Furstenberg-Szemerédi theorem and quasi-affine systems

“…In Section 2, we prove an odd ergodic Roth theorem, meaning a convergence theorem for progressions of length three restricted to a particular congruence class. The result follows easily from a theorem of Furstenberg [4] that explicitly describes the limit of expressions of the form…”

Difference sets and shifted primes

“…Let Q 1 = Q ∩ (0, 1] and Q 2 = Q∩ (1,2]. Let us assume without loss of generality that |Q 1 | ≥ (1/2)|Q|.…”

The structure of strongly stationary systems

“…The following result was proved in [Fu77] for ergodic systems and is needed in the sequel. The argument given there also works for nonergodic systems.…”

“…Furstenberg ([Fu77]) proved for ergodic systems that the k-step distal factor is characteristic for k + 1 terms. We want to use this result for general systems (not necessarily ergodic), so for completeness we include a proof that covers the general case.…”