Timeline for Are the Beatty primes asymptotically (Gowers) uniform?
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6 events
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| Aug 27, 2016 at 19:32 | comment | added | Terry Tao | 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.) | |
| Aug 27, 2016 at 18:53 | comment | added | Joel Moreira | 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 $f-g\leq 1_B\leq f+g$. | |
| Aug 25, 2016 at 19:43 | comment | added | Terry Tao | 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. | |
| Aug 25, 2016 at 11:20 | comment | added | Joel Moreira | @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 $F$. | |
| Aug 25, 2016 at 0:37 | comment | added | Terry Tao | 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 ). | |
| Aug 24, 2016 at 21:43 | history | asked | Joel Moreira | CC BY-SA 3.0 |