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A result of Green and Tao (initially conditional on two conjectures which were eventually settled by them and Ziegler) states that for any $s\in\mathbb N$, $$\lim_{w\to\infty}\limsup_{N\to\infty}\sup_{b\leq W,~ (b,W)=1}\|\Lambda(Wn+b)-1\|_{U^s_{[N]}}=0$$ where $W$ is the product of the first $w$ primes, $\Lambda$ is the von Mangoldt function, $\|\cdot\|_{U^s_{[N]}}$ denotes the Gowers $s$-norm and $n$ is the dummy variable of the function $n\mapsto\Lambda(Wn+b)-1$ inside the Gowers norm.

This result can be understood as saying that, after getting rid of local obstructions, the von Mangoldt function (and hence, in some sense, also the primes) is (Gowers) uniform. It has been used to find several patterns in the primes.

The question is whether a similar uniformity is known for the Beatty primes, i.e. primes of the form $\lfloor\theta n+\gamma\rfloor$ for fixed $\theta,\gamma\in\mathbb R$, $\theta>1$. More precisely:

Question: Is it known that for any $\theta,\gamma\in\mathbb R$ with $\theta>1$ irrational and any $s\in\mathbb N$ we have $$\lim_{w\to\infty}\limsup_{N\to\infty}\sup_{b\leq W,~ (b,W)=1}\Big\|1_B\cdot\big(\Lambda(Wn+b)-1\big)\Big\|_{U^s_{[N]}}=0,$$ where $W$ is the product of the first $w$ primes and $1_B$ is the indicator function of the set $B:=\big\{n\in\mathbb N:Wn+b\in\{\lfloor\theta m+\gamma\rfloor:m\in\mathbb Z\}\big\}$?

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    $\begingroup$ I think this will follow from our previous result by approximating $1_B$ by a trigonometric polynomial in $(Wn+b)/\theta$ which can then be more or less absorbed into the $U^s_{[N]}$ norm for any $s \geq 2$. More generally any weight that can be approximated by a nilsequence should be OK (see e.g. Corollary 2.2 of arxiv.org/pdf/1601.00562v3.pdf ). $\endgroup$
    – Terry Tao
    Commented Aug 25, 2016 at 0:37
  • $\begingroup$ @TerryTao I thought about approximating $1_B$ by polynomials, but the approximation (in the Besicovitch seminorm) does not seem good enough, specially considering that the von Mangoldt function is unbounded. Another way to think about it: one can read $1_B$ of a (1-step) nilsystem but with a discontinuous (yet Riemann integrable) function $F$. $\endgroup$
    – Joel Moreira
    Commented Aug 25, 2016 at 11:20
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    $\begingroup$ If one approximates $1_B = f + O(g)$ where $f,g$ are 1-step nilsystems with $g$ small and nonnegative, then by the triangle inequality one can estimate $\|1_B (\Lambda_{W,b}-1)\|_{U^s}$ by the sum of $\|f (\Lambda_{W,b}-1) \|_{U^s}$ and $O( \| g (\Lambda_{W,b}+1) \|_{U^s} )$. The former is small by Tanja's Corollary 2.2. The latter can be split into $O( \|g\|_{U^s} )$ and $O( \| g (\Lambda_{W,b}-1) \|_{U^s} )$; the second term is again small, and the first term can be controlled by a suitable $L^p$ norm of $g$ and will also be small. $\endgroup$
    – Terry Tao
    Commented Aug 25, 2016 at 19:43
  • $\begingroup$ 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 $f-g\leq 1_B\leq f+g$. $\endgroup$
    – Joel Moreira
    Commented Aug 27, 2016 at 18:53
  • $\begingroup$ Fair enough. The machinery in arxiv.org/pdf/math/0606088.pdf should eventually give this, though one may have to repeat quite a few of the arguments in that paper rather than just citing its main theorems as black boxes. (But the Chapter 11 machinery together with some clever use of the Cauchy-Schwarz-Gowers inequality may be enough to control $\|f(\Lambda_{W,b}-1)\|_{U^s}$ in terms of $\| \Lambda_{W,b}-1 \|_{U^s}$ plus small errors.) $\endgroup$
    – Terry Tao
    Commented Aug 27, 2016 at 19:32

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