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@StefanKohl: Ha, you're right! I convinced myself I could take $x\equiv -b/(a-1)\bmod d$ to check admissibility, but of course that fails in general when $d|a-1$... I still think it works if $a-1$ divides $b$, for instance with $x\mapsto 3x+2$.
Expanding a bit on my previous comment: for any affine map $\phi(x)=ax+b$ where $a\neq1$ and $a$ and $b$ are coprime, and for every $M$, the tuple $\{x,\phi(x),\phi^2(x),\dots,\phi^M(x)\}$ is admissible, and hence the Hardy-Littlewood prime tuple conjecture (a.k.a. Dickson's conjecture) predicts that there exists $x$ with all $\phi^k(x)$ prime, $0\leq k\leq M$. This implies that there are arbitrarily large connected components in $\Gamma$ if and only if some pair $(a_i,b_i)$ is co-prime with $a\neq1$.
@GerardoArizmendi Assuming the Dickson conjecture (en.wikipedia.org/wiki/Dickson%27s_conjecture) we should expect arbitrarily large components even when $k=1$. For instance, let $a_1=2,b_1=1$, then the statement that there exists a connected component with cardinality $M$ is equivalent to the existence of a prime $p$ such that $2^kp+2^k-1$ is prime for every $k=0,\dots,M$.
Just a remark: The union (call it $U$) of all connected components that are not singletons should have $0$ relative density within the primes: Denote by $\mathbb{P}$ the set of primes. For any fixed $a,b\in\mathbb{N}$, the set $\{p\in\mathbb{P}:ap+b\in\mathbb{P}\}$ should have $0$ relative density within the primes (I imagine this follows "easily" from sieve theory but I could only find references when $a=1$) and so a finite union of such sets still has relative density $0$. And $U$ is contained in this union.
In Problem 2 you're taking the limit as $N\to\infty$ on the left hand side, so it's strange to also have $N$ on the right. Perhaps you want to divide by $N^{1/k}$ on the left instead?
I was thinking of the version of Choquet's theorem which says that every point of a compact convex set ${\mathcal D}$ is the barycenter of a probability measure on the extreme points of ${\mathcal D}$, although I realize now that is not exactly the version on wikipedia. But the version on wikipedia is good enough to answer your question: Let $F:{\mathcal D}\to{\mathbb R}$ be the affine function $F:\mu\mapsto\int_{\mathbb R}f(x)d\mu(x)$. Then we have $F(\mu)=\int_{\mathcal E}F(\rho)d\nu(\rho)\leq\sup_{\rho\in{\mathcal E}}F(\rho)$.
Thanks for the example and the reference. This does answer the first question, but I'm not sure the Morse system is distal. In fact I thought no symbolic system can be distal (unless it is finite). But perhaps one can project onto the largest distal factor and still have the same property?
Thanks for the answer but I didn't understand something: it seems to me that the intervals $U_N$ could shrink so that the intersection will be a single point at most. How do we guarantee we can fit a Cantor set in there?
This seems to work, thanks! However, Tanja's corollary only gives orthogonality to nilsequences; one still needs to invoke some form of the inverse theorem for Gowers norms and deal with the fact that $\Lambda_{W,b}$ is unbounded (albeit uniform). Btw, I didn't know the notation $1_B=f+O(g)$, but interpreted it as $f-g\leq 1_B\leq f+g$.
