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A contribution to infinite disjoint covering systems

János Barát, Péter P. Varjú (2005)

Journal de Théorie des Nombres de Bordeaux

Let the collection of arithmetic sequences { d i n + b i : n } i I be a disjoint covering system of the integers. We prove that if d i = p k q l for some primes p , q and integers k , l 0 , then there is a j i such that d i | d j . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to 1 . The above conjecture holds for saturated systems with d i such that the product of its prime factors is at most 1254 .

A Freĭman-type theorem for locally compact abelian groups

Tom Sanders (2009)

Annales de l’institut Fourier

Suppose that G is a locally compact abelian group with a Haar measure μ . The δ -ball B δ of a continuous translation invariant pseudo-metric is called d -dimensional if μ ( B 2 δ ) 2 d μ ( B δ ) for all δ ( 0 , δ ] . We show that if A is a compact symmetric neighborhood of the identity with μ ( n A ) n d μ ( A ) for all n d log d , then A is contained in an O ( d log 3 d ) -dimensional ball, B , of positive radius in some continuous translation invariant pseudo-metric and μ ( B ) exp ( O ( d log d ) ) μ ( A ) .

A Note on squares in arithmetic progressions, II

Enrico Bombieri, Umberto Zannier (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show that the number of squares in an arithmetic progression of length N is at most c 1 N 3 / 5 log N c 2 , for certain absolute positive constants c 1 , c 2 . This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent 2 3 in place of our 3 5 . The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus 5 as in [1].

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

A geometric progression of length k and integer ratio is a set of numbers of the form a , a r , . . . , a r k - 1 for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ( a i ) i = 1 of positive real numbers with a₁ = 1 such that the set G ( k ) = i = 1 ( a 2 i , a 2 i - 1 ] contains no geometric progression of length k and integer ratio. Moreover, G ( k ) is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...

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