Combinatorics, Probability and Computing
Towards Dimension Expanders over Finite Fields
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Explicit constructions of linear size superconcentrators
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Euclidean Sections of $\ell_1^N$ with Sublinear Randomness and Error-Correction over the Reals
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Product set estimates for non-commutative groups
Combinatorica
Quantum Information & Computation
IEEE Transactions on Information Theory - Part 1
Proving expansion in three steps
ACM SIGACT News
| Hi-index | 0.00 |
This work presents an explicit construction of a family of monotone expanders, which are bi-partite expander graphs whose edge-set is defined by (partial) monotone functions. The family is essentially defined by the Mobius action of SL2(R), the group of 2 x 2 matrices with determinant one, on the interval [0,1]. No other proof-of-existence for monotone expanders is known, not even using the probabilistic method. The proof extends recent results on finite/compact groups to the non-compact scenario. Specifically, we show a product-growth theorem for SL2(R); roughly, that for every A ⊂ SL2(R) with certain properties, the size of AAA is much larger than that of A. We mention two applications of this construction: Dvir and Shpilka showed that it yields a construction of explicit dimension expanders, which are a generalization of standard expander graphs. Dvir and Wigderson proved that it yields the existence of explicit pushdown expanders, which are graphs that arise in Turing machine simulations.