Relative Property (T) for Topological Groups

Tao, Jicheng; Yan, Wen

doi:10.4236/am.2014.519285

Cited by 3 publications

(4 citation statements)

References 7 publications

(10 reference statements)

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“…Another thing to notice is that, for distinct distances, the situation is quite different when V is (or contains) a plane or a sphere, in which case the bound goes up to Ω(n/ log n) [39,67] (see also Sheffer's survey [61] for details).…”

Section: Remarkmentioning

confidence: 99%

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

Sharir¹,

Solomon²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study a wide spectrum of incidence problems involving points and curves or points and surfaces in R 3 . The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [38,39], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [40], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in R 3 .In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic curves, and those involving points and constant-degree algebraic surfaces. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space.The case of points and curves has been considered in several previous studies, starting with Guth and Katz's work on points and lines [39]. Our results, which are based on a recent work of Guth and Zahl [40] concerning surfaces that are doubly ruled by curves, provide a grand generalization of all previous results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways, recent bounds involving points and circles (in [54]), and points and arbitrary constant-degree algebraic curves (in [53]). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge.In the case of points and surfaces, the incidence graph between them can contain large complete bipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlier studies, we do not rule out this possibility, which makes our approach more general). Our bounds estimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our results provide a "grand generalization" of most of the previous studies of (special instances of) this problem.As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of n points in R 3 . We obtain the bound O(n 15/7 ), which improves the bound of Agarwal et al. [1].As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of n points in R 3 , two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in P 1 × P 2 , where P 1 is contained in a variety and P 2 is arbitrary.Recently, Sharir and Zahl [60] have considered general families of con...

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Section: Remarkmentioning

confidence: 99%

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

Sharir¹,

Solomon²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…We also observe that In particular X/H is a regular graph. Observe that by (19), there are |H| distinct right cosets in HaH. Therefore, X/H is regular of degree |H|.…”

Section: Reduction To the Case When |H| <mentioning

confidence: 99%

“…However, deriving a similar result in the general nonabelian case has not been attempted until recently, when the topic has attracted the attention of a number of researchers in the area of so-called approximate groups, see e.g. [19].…”

Section: Introductionmentioning

confidence: 99%

A structure theorem for small sumsets in nonabelian groups

Serra

Zémor

2013

European Journal of Combinatorics

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Let G be an arbitrary finite group and let S and T be two subsets such that |S| ≥ 2, |T | ≥ 2, and |T S| ≤ |T | + |S| − 1 ≤ |G| − 2. We show that if |S| ≤ |G| − 4|G| 1/2 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS| ≤ |S| + |H| − 1 or |SH| ≤ |S| + |H| − 1. This extends to the nonabelian case classical reults for abelian groups. When we remove the hypothesis |S| ≤ |G| − 4|G| 1/2 we show the existence of counterexamples to the above characterization whose structure is described precisely.

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“…Constructive consequences. In [4], Tao alludes to the fact that arguments in combinatorics involving idempotent ultrafilters are highly nonconstructive as one needs to use the axiom of choice multiple times to prove the existence of an idempotent ultrafilter. (As a curiosity, it is not currently known whether or not the existence of idempotent ultrafilters on N is equivalent, over ZF, to the existence of nonprincipal ultrafilters on N.) In relation to this fact, here is a vague conjecture:…”

Section: Introductionmentioning

confidence: 99%