Is the structure $(\omega,+,2^n)$ undecidable?

Let $L$ be the first-order language with binary function symbol $+$, unary function symbol $E$. Let $T$ be the set of sentences over this language that are true in the natural numbers, with $+$ interpreted in the usual way, and $E(n$) interpreted as $2^n$. Is the set $T$ recursive?

-
Also asked here: math.stackexchange.com/questions/349383/… –  Andres Caicedo Apr 2 '13 at 22:04
Thank you for the help! –  danulg Apr 3 '13 at 11:32
As Marty explained in this answer, this question is the central topic of the paper On the expansion $\langle \mathbb{N},+,2^x\rangle$ of Presburger arithmetic, by Françoise Point, based on a joint proceedings paper with G. Cherlin on a result credited to Alexei L. Semenov.