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We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding whether, given p $\in$ N, there exists such a constant subsequence along an arithmetic progression of common difference p. In the special case of uniformly recurrent automatic sequences we explicitely describe the sets of such p by means of automata.

Decidability, Arithmetic Subsequences and Eigenvalues of Morphic Subshifts