TITLE: Are these sets stationary?
QUESTION [6 upvotes]: Assume $\kappa > \omega$ is a cardinal of uncountable cofinality and $S$ is a stationary set in $\kappa$. If $\alpha < \kappa$ is a successor ordinal then the set $S+\alpha=\{\sigma+\alpha : \sigma \in S\}$ is not stationary in $\kappa$ because it does not intersect the subset of limit ordinals in $\kappa$ (which is club in $\kappa$). Is $S + \alpha$ stationary in $\kappa$ if $\alpha < \kappa$ is a limit ordinal? What about the set $\{\sigma + \sigma : \sigma \in S\}$?

REPLY [5 votes]: If $\kappa$ is regular and uncountable, then no set of the form $S = A + \alpha$ can be stationary (for any $A \subseteq \kappa$ and $\alpha \in \kappa$) by Fodor's lemma.  The function which sends each $\beta \in S$ to the least $\gamma$ such that $\gamma + \alpha = \beta$ is regressive.
A similar argument shows that no set of the form $\{\alpha + \alpha : \alpha \in A\}$ is stationary (again, for regular $\kappa$).