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Families of plane curves having translates in a set of measure zero
Author(s) -
Sawyer Eric
Publication year - 1987
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300013292
Subject(s) - mathematics , differentiable function , measure (data warehouse) , zero (linguistics) , lipschitz continuity , null set , plane (geometry) , plane curve , function (biology) , mathematical analysis , line (geometry) , set (abstract data type) , order (exchange) , pure mathematics , combinatorics , geometry , linguistics , evolutionary biology , biology , finance , economics , database , computer science , programming language , philosophy
We construct a universal function φ on the real line such that, for every continuously differentiable function f the range of f – φ has measure zero. We then apply this to obtain results on curve packing that generalize the Besicovitch set. In particular, we show that given a continuously differentiable family of measurable curves, there exists a plane set of measure zero containing a translate of each curve in the family. Examples are given to show that the differentiability hypothesis cannot be weakened to a Lipschitz condition of order α for any 0<α<1.