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We consider Presburger arithmetic extended by the sine function, call thisextension sine-Presburger arithmetic ($\sin$-PA), and systematically studydecision problems for sets of sentences in $\sin$-PA. In particular, we detaila decision algorithm for existential $\sin$-PA sentences under assumption ofSchanuel's conjecture. This procedure reduces decisions to the theory of theordered additive group of real numbers extended by sine, which is decidableunder Schanuel's conjecture. On the other hand, we prove that four alternatingquantifier blocks suffice for undecidability of $\sin$-PA sentences. To do so,we explicitly interpret the weak monadic second-order theory of the grid, whichis undecidable, in $\sin$-PA.

Decidability bounds for Presburger arithmetic extended by sine