Research Images as Art / Art Images as Research 2010/11
Jack Grahl, Mathematics
Trying to sketch the optimal exceptional set
This image is a composite of sketches that I made while trying to solve the following problem: What is the smallest number d such that for every Lebesgue measurable set in the line, there do not exist any points at which the set has Lebesgue density between d and 1-d? This question deals with arrangements of points along a line, and for simplicity one can look at a related but equivalent question, where all the arrangements are made up of disjoint intervals or line segments. The sketches attempt to represent the features that the 'best possible' arrangements of line segments might have. The question was solved earlier this year by another researcher from Charles University in Prague who proved that the minimal value of d is 0.268486884…
Produced: Drawing with digital manipulation.
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