dc.contributor.author | Liu, Yu-Ru | |
dc.contributor.author | Spencer, Craig V. | |
dc.contributor.author | Zhao, Xiaomei | |
dc.date.accessioned | 2023-10-03 15:11:57 (GMT) | |
dc.date.available | 2023-10-03 15:11:57 (GMT) | |
dc.date.issued | 2010 | |
dc.identifier.uri | https://doi.org/10.4064/aa142-4-6 | |
dc.identifier.uri | http://hdl.handle.net/10012/20001 | |
dc.description.abstract | 1. Introduction. For r, s ∈ N = {1, 2, . . .} with s ≥ 2r + 1, let (bi,j ) be
an r×s matrix whose elements are integers. Suppose that bi,1+· · ·+bi,s = 0
(1 ≤ i ≤ r). Suppose further that among the columns of the matrix, there
exist r linearly independent columns such that, if any of the r columns are
removed, the remaining n − 1 columns of the matrix can be divided into
two sets so that among the columns of each set there are r linearly independent columns. For k ∈ N, denote by D([1, k]) the maximal cardinality
of an integer set A ⊆ [1, k] such that the equations bi,1x1 + · · · + bi,sxs = 0
(1 ≤ i ≤ r) are never satisfied simultaneously by distinct elements x1, . . . , xs
∈ A. Using techniques similar to his work on sets free of three-term arithmetic progressions (see [4]), Roth [5] showed that
D([1, k]) k/(log log k)
1/r2
.
In this paper, we will build upon the methods in [2] to study an analogous
question in function fields | en |
dc.language.iso | en | en |
dc.publisher | Institute of Mathematics | en |
dc.relation.ispartofseries | Acta Arithmetica;142 | |
dc.rights | Attribution 4.0 International | * |
dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | * |
dc.subject | Roth's theorem | en |
dc.subject | function fields | en |
dc.subject | circle method | en |
dc.title | Roth's theorem on systems of linear forms in function fields | en |
dc.type | Article | en |
dcterms.bibliographicCitation | Liu, Y.-R., Spencer, C. V., & Zhao, X. (2010). Roth’s theorem on systems of linear forms in function fields. Acta Arithmetica, 142(4), 377–386. https://doi.org/10.4064/aa142-4-6 | en |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.contributor.affiliation2 | Pure Mathematics | en |
uws.typeOfResource | Text | en |
uws.peerReviewStatus | Reviewed | en |
uws.scholarLevel | Faculty | en |