Contractions and expansion
Let A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lie...
| Main Authors: | , |
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| Format: | Journal article |
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2011
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| _version_ | 1797080051298074624 |
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| author | Breuillard, E Green, B |
| author_facet | Breuillard, E Green, B |
| author_sort | Breuillard, E |
| collection | OXFORD |
| description | Let A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lies between a and x. Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh. |
| first_indexed | 2024-03-07T00:54:35Z |
| format | Journal article |
| id | oxford-uuid:879843c1-33c6-4eff-b6f5-5189a326d95c |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:54:35Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:879843c1-33c6-4eff-b6f5-5189a326d95c2022-03-26T22:11:42ZContractions and expansionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:879843c1-33c6-4eff-b6f5-5189a326d95cSymplectic Elements at Oxford2011Breuillard, EGreen, BLet A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lies between a and x. Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh. |
| spellingShingle | Breuillard, E Green, B Contractions and expansion |
| title | Contractions and expansion |
| title_full | Contractions and expansion |
| title_fullStr | Contractions and expansion |
| title_full_unstemmed | Contractions and expansion |
| title_short | Contractions and expansion |
| title_sort | contractions and expansion |
| work_keys_str_mv | AT breuillarde contractionsandexpansion AT greenb contractionsandexpansion |