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New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia (2015)

Colloquium Mathematicae

Let p p ¯ ( n ) denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, p p ¯ ( 3 n + 2 ) 0 ( m o d 3 ) . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for p p ¯ ( n ) . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for p p ¯ ( n ) . Furthermore, they also constructed infinite families of congruences for p p ¯ ( n ) modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite...

Non-degenerate Hilbert cubes in random sets

Csaba Sándor (2007)

Journal de Théorie des Nombres de Bordeaux

A slight modification of the proof of Szemerédi’s cube lemma gives that if a set S [ 1 , n ] satisfies | S | n 2 , then S must contain a non-degenerate Hilbert cube of dimension log 2 log 2 n - 3 . In this paper we prove that in a random set S determined by Pr { s S } = 1 2 for 1 s n , the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly log 2 log 2 n + log 2 log 2 log 2 n and determine the threshold function for a non-degenerate k -cube.

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