The inverse problem for representation functions of additive bases
Melvyn B. Nathanson
Abstract
Let A be a set of integers. For every integer n, let rA,2(n) denote the number of representations of n in the form n = a1 + a2, where a1 and a2 are in A and a1 ≤ a2. The function rA,2: Z N0 ∞ is the representation function of order 2 for A. The set A is called an asymptotic basis of order 2 if rA,2-1(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of two not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z N0 ∞ is any function such that f-1(0) is finite, then there exists a set A of integers such that f(n) = rA,2(n) for all n ∈ . Moreover, the set A can be constructed so that carda∈ A : |a| ≤ x x1/3.
Create a lesson
Related papers
On exponential sums with multiplicative coefficients
Haozhe Gou
Automorphic Dirac Operators and an Analogue of the Quantum Harmonic Oscillator on Γ H
Marcella Manivel
Variation of Shafarevich-Tate groups in Hida families
Enrico Da Ronche, Matteo Longo, Stefano Vigni et al.
The discrete second moment of the Riemann zeta-function along powers of Gram points
Marius Dundulis, Ramūnas Garunkštis, Antanas Laurinčikas et al.
Logarithmic exponential sums with multiplicative coefficients
Haozhe Gou
Ekedahl-Oort strata with a superspecial factor in unitary Shimura varieties
Emerald Andrews, Deewang Bhamidipati, Heidi Goodson et al.