Displaying similar documents to “On the lonely runner conjecture”

Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov (2014)

Acta Arithmetica

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Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound O ( n 3 / 2 ) .

On non-intersecting arithmetic progressions

Régis de la Bretèche, Kevin Ford, Joseph Vandehey (2013)

Acta Arithmetica

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We improve known bounds for the maximum number of pairwise disjoint arithmetic progressions using distinct moduli less than x. We close the gap between upper and lower bounds even further under the assumption of a conjecture from combinatorics about Δ-systems (also known as sunflowers).

Addendum to: On volumes of arithmetic quotients of SO (1, n)

Mikhail Belolipetsky (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.