Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period p n + 1 , where p = d R + 1 is an odd prime, d , R are two non-negative integers, and n > 0 is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period p n + 1 ( n ≥ 0 ) with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

Simultaneous stationary reflection and square sequences

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

Forcing Closed Unbounded Subsets of אω1+1

Using square sequences, a stationary subset ST of אω1+1 is constructed from a tree T of height ω1, uniformly in T. Under suitable hypotheses, adding a closed unbounded subset to ST requires adding a cofinal branch to T or collapsing at least one of ω1, אω1, and אω1+1. An application is that in ZFC there is no parameter free definition of the family of subsets of אω1+1 that have a closed unbounded subset in some ω1, אω1, and אω1+1 preserving outer model.

Weak Compactness and No Partial Squares

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.