As we know there are quantum analogue of tori called quantum tori generated by noncommuting operators $(A_1,\dots,A_n)$ with $A _iA_j=A_jA_ie^{2\pi i\alpha}$ where $\alpha$ is a irrational number as a universal $C^*$ algebra and a similar definition of a quantum Euclidean space, can we define a notion of quantum integer space, which completes the circle of classical groups? Also can we have a notion of quantum $p$-adic group? Also can we define some sort of Fourier transform on these spaces, as we can do for quantum tori and quantum Euclidean spaces?

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I'm not sure what you mean by "quantum integer space", but if you're asking for an analog of the Fourier correspondence between $\mathbb{T}^2$ and $\mathbb{Z}^2$ I think the answer is no. We do have a notion of duality for quantum groups, but there is no reasonable way to put a quantum group structure on the quantum torus.