46 research outputs found
On the restriction problem for discrete paraboloid in lower dimension
We apply geometric incidence estimates in positive characteristic to prove
the optimal Fourier extension estimate for the paraboloid in the
four-dimensional vector space over a prime residue field. In three dimensions,
when is not a square, we prove an extension
estimate, improving the previously known exponent Comment: Final versio
Difference sets are not multiplicatively closed
Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp.
The famous sum-product problem of Erdős and Szemerédi asks the following. Let be a set of real numbers. Define the _sumset_ of to be the set and the _product set_ to be the set . Must at least one of and have size ? Since neither set can have size greater than , this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was , up to logarithmic factors [3], though the exponent has now been very slightly improved [2].
More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if is a subset of a finite field and , then one of and must have size at least for some that depends on only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that does not.
This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if has a large difference set , it is perfectly possible for to be much bigger than . Nevertheless, when is a set of real numbers, it still implies, as the main result of the paper shows, that the product set has size at least , where is an absolute constant. The same is true of the quotient set .
The paper also proves a result of a similar kind in the finite field provided that is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup of that is not too large is not the difference set of any subset of . (The precise assumption required is that has order at most .)
[1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343)
[2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771)
[3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040)
[Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)<sup