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Problems in additive number theory, II: Linear forms and complementing sets

Melvyn B. Nathanson — 2009

Journal de Théorie des Nombres de Bordeaux

Let ϕ ( x 1 , ... , x h , y ) = u 1 x 1 + ⋯ + u h x h + v y be a linear form with nonzero integer coefficients u 1 , ... , u h , v . Let 𝒜 = ( A 1 , ... , A h ) be an h -tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form ϕ and the sets 𝒜 and B as follows : R 𝒜 , B ( ϕ ) ( n ) = card { ( a 1 , ... , a h , b ) ∈ A 1 × ⋯ × A h × B : ϕ ( a 1 , ... , a h , b ) = n } . If this representation function is constant, then the set B is periodic and the period of B will be bounded in terms of the diameter of the finite set { ϕ ( a 1 , ... , a h , 0 ) : ( a 1 , ... , a h ) ∈ A 1 × ⋯ × A h } . Other results for complementing sets with respect to linear forms are also proved....

Polynomial growth of sumsets in abelian semigroups

Melvyn B. Nathanson;  Imre Z. Ruzsa — 2002

Journal de théorie des nombres de Bordeaux

Let S be an abelian semigroup, and A a finite subset of S . The sumset h A consists of all sums of h elements of A , with repetitions allowed. Let | h A | denote the cardinality of h A . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p ( t ) such that | h A | = p ( h ) for all sufficiently large h . Lattice point counting is also used to prove that sumsets of the form h 1 A 1 + ⋯ + h r A r have multivariate polynomial growth.

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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