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We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form  h ( n ) with   $n\in \mathbb {N}$   or  h ( p ) with  p  prime for appropriate   $h\in \mathbb {Z}[x]$  . The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik’s theorem.

Polynomials and Primes in Generalized Arithmetic Progressions