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English is not my native tongue. German is not my native tongue. French is not my native tongue.
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Boundedness of solutions of a difference equation
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Boundedness of solutions of a difference equation
@RobertIsrael: I see. I updated my response accordingly, and added a comment below the original post. BTW the question in the original post looked quite random to me, and I certainly did not pay enough attention to it. 
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Boundedness of solutions of a difference equation
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Boundedness of solutions of a difference equation
As @RobertIsrael pointed out to me, the problem asked above is Conjecture 8 in the paper anubih.ba/Journals/vol.8,no2,y12/11LadasLugoPalladino.pdf . My answer below was accepted by the OP, but unfortunately it answered a much simpler question (due to my misunderstanding of the original question). Ideally I would delete my answer, but I cannot, because it is protected now. 
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Boundedness of solutions of a difference equation
@RobertIsrael: Thanks for pointing this out. I agree that from a formal point of view your interpretation is the correct one, and there is more work to do. On the other hand, I tend to believe that the OP meant the problem how I understood it, otherwise he/she would not have accepted my answer. Feel free to add your response to clarify the situation! 
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awarded  Enlightened 
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awarded  Nice Answer 
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Question about zeta function of function field in 1 variable over $\mathbb{F}_q$
@KConrad: Thank you. I missed your note "esp. Section 5.2.2". BTW I did not know about Roquette's paper until your comment today, and now that I have read parts of it, I find it highly interesting and entertaining. 
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Question about zeta function of function field in 1 variable over $\mathbb{F}_q$
@KConrad: I agree that the question is basic and can be answered after minimal selfstudy. On the other hand, I would like to defend the OP, who is probably a student, in that only a few hours passed since she/he received from you the link to Roquette's paper, and the quoted formula is on page 49. 
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Question about zeta function of function field in 1 variable over $\mathbb{F}_q$
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answered  Question about zeta function of function field in 1 variable over $\mathbb{F}_q$ 
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answered  Reference request, zeta function is rational function via RiemannRoch? 
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Radial limit does not exist almost everywhere
@MattYoung: Thank you! I am glad you scanned the paper and posted to your webpage. I read it quickly (skipping some details), and found it very nice. 
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Analytic continuation of intertwining operator
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Analytic continuation of intertwining operator
I think this is nontrivial, e.g. one has to look at the Jacquet modules of the reducible principal series representations $\mathbf{H}(\eta_v)$ and $\mathbf{H}(\tilde\eta_v)$. See Theorem 8.12.22 in GoldfeldHundley's Volume I, and also Exercise 4.5.5 in Bump: Automorphic forms and representations. 
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How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
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How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
The "solenoid structure" you are asking about is addressed in the last sentence of my response. Briefly, $\mathrm{PGL}_2(\mathbb{Q})\backslash\mathrm{PGL}_2(\mathbb{A})$ can be identified with the inverse limit of $\Gamma(N)\backslash\mathrm{PGL}_2(\mathbb{R})$, where $\Gamma(N)$ is the usual principal congruence subgroup modulo $N$. 
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How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
@johnmangual: The limit that you mention in your comment above is an inverse limit of topological groups, a standard construction. That is, the inverse limit has the structure of a topological group by definition. See en.wikipedia.org/wiki/Inverse_limit and math.stackexchange.com/questions/711334/… 
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How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
@johnmangual: For comparison, $\mathbb{R}^2$ divided by $\{(x,x):\ x\in\mathbb{Z}\}$ is very different from $(\mathbb{R}/\mathbb{Z})^2$. In the first case you get a group isomorphic to $\mathbb{R}\times(\mathbb{R}/\mathbb{Z})$. 
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How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
@johnmangual: What you write is not the adelic quotient. You have to divide the restricted product $\mathrm{PGL}_2(\mathbb{A})$ instead of taking a restricted product of the local quotients (which are not even Hausdorff). Note also that $\mathbb{R}$ is usually denoted by $\mathbb{Q}_\infty$ instead of $\mathbb{Q}_{1}$. 