Timeline for Is the sumset or the sumset of the square set always large?
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12 events
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| Oct 8, 2015 at 17:12 | comment | added | Boris Bukh | @BrendanMurphy Indeed, over $\mathbb{R}$ the problem of $A^2+A^2$ vs $A^3+A^3$ is settled by the result of Elekes-Nathanson-Ruzsa. The question is over $\mathbb{F}_p$. | |
| Oct 8, 2015 at 16:20 | comment | added | Brendan Murphy | @OliverRoche-Newton Since the method of Elekes-Nathanson-Ruzsa gives a lower bound for $\max\{|B+B|, |B^{3/2}+B^{3/2}\}$, taking $B=A^2$ might say something about the question of $A^2+A^2$ vs $A^3+A^3$ (over $\mathbb{R}$ at least). | |
| Sep 28, 2015 at 4:53 | comment | added | Oliver Roche-Newton | @BorisBukh, I suspect that some recent progress for incidence geometry in the complex setting means that one can now get similar bounds for the complex case as those known for the reals. This paper of Sheffer and Zahl (arxiv.org/abs/1502.07003) gives some generalisations of the Szemeredi-Trotter Theorem for nice curves in the complex plane. I would need to check, but it seems likely that this could be combined with the arguments of Elekes-Nathanson-Ruzsa to give a decent estimate for $\max \{|A+A|,|A^2+A^2| \} $. | |
| Sep 6, 2015 at 20:43 | comment | added | Terry Tao | It would be good to have the Rudnev and Roche-Newton-Rudnev-Shkredov papers read through carefully; their method looks quite promising if their calculations are correct (but there were some issues with previous versions of these manuscripts). | |
| Sep 6, 2015 at 17:41 | comment | added | Boris Bukh | @MarkLewko I admit of having missed this paper. I retract my statement of "one proof technique" until I read the paper (and the paper of Rudnev on which it is based). | |
| Sep 6, 2015 at 17:35 | vote | accept | Mark Lewko | ||
| Sep 6, 2015 at 17:28 | comment | added | Mark Lewko | I'm not sure I'd agree we only have one proof technique now. See the recent paper of Roche-Newton, Rudnev and Shkredov which derives sum-product estimates from an incidence bound which is proved via the polynomial method. This strikes me as quite different than the initial approach. My application was to try to improve / give an alternate / more robust / more general approach to some results which are based on sum-product estimates. It might be interesting to check if expansion estimates of this form could be used to deduce incidence estimates for curves which I believe is open in F_p. | |
| Sep 6, 2015 at 17:26 | comment | added | Boris Bukh | @MarkLewko The result in $\mathbb{R}$ is available with better exponents. See Corollary 3.2 in ams.org/mathscinet-getitem?mr=1772612. By the general algebro-geometric nonsense, a bound in $\mathbb{F}_p$ automatically implies the same bound in $\mathbb{C}$. | |
| Sep 6, 2015 at 17:17 | comment | added | Boris Bukh | @MarkLewko The problem with the exponents is that we have exactly one proof technique for proving sum-product-type estimates for small sets in finite fields, and this proof technique is quite wasteful. However, I do not know of any applications where the numeric exponents make qualitative difference (unless one somehow would able to prove the sharp exponents, which we currently cannot). If in your application the exponents do matter, I would be very interested. | |
| Sep 6, 2015 at 17:11 | comment | added | Mark Lewko | Thank you, Boris! This addresses the finite field case. Unfortunately, the exponents are worse than what is currently known for the sum-product problem, however. | |
| Sep 6, 2015 at 16:42 | history | edited | Boris Bukh | CC BY-SA 3.0 |
added 9 characters in body
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| Sep 6, 2015 at 16:37 | history | answered | Boris Bukh | CC BY-SA 3.0 |