Timeline for Additive energy of Piatetski-Shapiro sequences
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2015 at 12:29 | vote | accept | Kurisuto Asutora | ||
| Oct 22, 2015 at 2:33 | comment | added | Brendan Murphy | @MarkLewko I think the multiplicities of those curves can be quite high; eg the mean value theorem shows that the number of solutions to $x-1<(n+s)^c - n^c < x +1$ is $\ll 1 + N^{2-c}/s$ (I might be missing a factor of $\log N$). Still, it would be interesting to attack these problems from a geometric angle. | |
| Oct 22, 2015 at 2:25 | answer | added | Brendan Murphy | timeline score: 0 | |
| Oct 21, 2015 at 22:11 | answer | added | Eric Naslund | timeline score: 1 | |
| Oct 20, 2015 at 21:18 | comment | added | Mark Lewko | Using Elekes' relation between sum-product and incidence theorems (see the sketch in my answer here mathoverflow.net/questions/217557/…), this should follow from a Szemeredi-Trotter theorem for translates of the curve $(x, \lfloor x^{c} \rfloor )$. However the crossing number proof of Szemeredi-Trotter only uses that translates of this curve intersects in $O(1)$ places and should apply here. | |
| Oct 20, 2015 at 21:16 | comment | added | Mark Lewko | I haven't checked this carefully but here is an alternate approach that seems simpler. By BSG / "routine" additive combinatorics it suffices to show that $|A+A| > |A|^{1+c}$ for some fixed $c>0$ and $A$ a large subset of $[N]$ (that is $|A|>N^{1-a}$ for some small $a$). It thus should suffice to show that $\max(|A+A|, |f(A)+f(A)|) > |A|^{1+c}$ for $c>0$ where $f(x) = \lfloor x^{c} \rfloor$. | |
| Oct 20, 2015 at 19:07 | comment | added | Mark Lewko | ... See the introduction of this paper "Exceptional set of a representation with fractional powers" by Balanzario, Garaev, and Zuazua for a discussion of this result. This isn't quite a proof since one needs to consider all restricted sumsets of positive density within $A + A$ and not just the full set, however one might be able to adapt the ideas from that proof or use the "restriction theorem" for the P-S sequence of Mirek to pass to the general case (arxiv.org/abs/1305.0043). | |
| Oct 20, 2015 at 19:07 | comment | added | Mark Lewko | I believe one should be able to prove this with existing technology. Let me give an incomplete sketch (which is in line with the ideas of previous comments). First by the Balog-Szemeredi-Gowers lemma if the claim is false then one would have that every "restricted" sumset of positive density in $A \times A$ has small doubling. That is $|A + A| \lesssim N^{1/c+o(1)}$ (where the sumset is restrict to a positive proportion of the set $A \times A$). On the other hand it is known for $c<3/2$ the sumset of the Piatetski-Shapiro sequence has positive density in the integers. ... | |
| Oct 20, 2015 at 18:01 | comment | added | Seva | The additive energy will be smaller, but I would expect not much smaller, than the number of quadruples $(n_1,n_2,n_3,n_4)$ with $|n_1^c+n_2^c-n_3^c-n_4^c|<2$. I guess it should be possible to estimate the total number of such quadruples using some analytic technique. | |
| Oct 20, 2015 at 18:00 | comment | added | Boris Bukh | You might be able to adapt the argument from arxiv.org/abs/math/0503069 to this situation (there are few distinct consecutive differences in this problem, as opposed to all distinct consecutive differences being distinct) | |
| Oct 20, 2015 at 17:48 | comment | added | Boris Bukh | I was too careless --- sorry. | |
| Oct 20, 2015 at 17:27 | comment | added | Kurisuto Asutora | To apply Corollary 3.2 from this paper it is necessary to have strict convexity (noted before the statement of Theorem 1) - which we do not have in the present case, since convexity is ruined by the floor-function. | |
| Oct 20, 2015 at 15:46 | comment | added | Boris Bukh | See Corollary 3.2 in ams.org/mathscinet-getitem?mr=1772612 for set version of the statement that you want. You can then use Balog-Szemeredi-Gowers to deduce the energy version. | |
| Oct 20, 2015 at 14:08 | history | edited | Kurisuto Asutora |
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| Oct 20, 2015 at 12:17 | history | asked | Kurisuto Asutora | CC BY-SA 3.0 |