Oriol Serra, Gilles Zémor (2009)
Annales de l’institut Fourier
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We prove that there is a small but fixed positive integer such that for every prime larger than a fixed integer, every subset of the integers modulo which satisfies and is contained in an arithmetic progression of length . This is the first result of this nature which places no unnecessary restrictions on the size of .
Imre Z. Ruzsa (1991)
Acta Arithmetica
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1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length . Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1). A set of residues can be used to get a set of integers in an obvious...
Zoltán Daróczy, Gyula Maksa (1999)
Colloquium Mathematicae
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We solve Matkowski's problem for strictly comparable quasi-arithmetic means.
Julien Cassaigne, Sébastien Ferenczi, Christian Mauduit, Joël Rivat, András Sárközy (2000)
Acta Arithmetica
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Schlage-Puchta, Jan-Christoph (2003)
Journal of Integer Sequences [electronic only]
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Maurizio Laporta (2012)
Journal de Théorie des Nombres de Bordeaux
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We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer in the form with prime numbers such that , under suitable hypothesis on for every integer .
Florian Luca (2012)
Communications in Mathematics
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We answer a question of Bednarek proposed at the 9th Polish, Slovak and Czech conference in Number Theory.