On rich lines in grids
| dc.creator | Borenstein, Evan | |
| dc.creator | Croot, Ernie | |
| dc.date | 2008-07-15 | |
| dc.date.accessioned | 2026-07-07T09:50:30Z | |
| dc.date.available | 2026-07-07T09:50:30Z | |
| dc.description | In this paper we show that if one has a grid A x B, where A and B are sets of n real numbers, then there can be only very few ``rich'' lines in certain quite small families. Indeed, we show that if the family has lines taking on n^epsilon distinct slopes, and where each line is parallel to n^epsilon others (so, at least n^(2 epsilon) lines in total), then at least one of these lines must fail to be ``rich''. This result immediately implies non-trivial sum-product inequalities; though, our proof makes use of the Szemeredi-Trotter inequality, which Elekes used in his argument for lower bounds on |C+C| + |C.C|. | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/0807.2420 | |
| dc.identifier | http://arxiv.org/abs/0807.2420 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164963 | |
| dc.subject | Combinatorics | |
| dc.subject | 05D99 | |
| dc.title | On rich lines in grids | |
| dc.type | text |