34 citations · 268 across the 120 of their papers we have counts for
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Inverse Problems for Representation Functions in Additive Number Theory
Melvyn B. Nathanson
For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of…
Asymptotic estimates for phi functions for subsets of {m+1, m+2,...,n}
Melvyn B. Nathanson, Brooke Orosz
Let f(m,n) denote the number of relatively prime subsets of {m+1,m+2,...,n}, and let Phi(m,n) denote the number of subsets A of {m+1,m+2,...,n} such that gcd(A) is relatively prime…
Representation functions of bases for binary linear forms
Melvyn B. Nathanson
Let F(x_1,...,x_m) = u_1 x_1 + ... + u_m x_m be a linear form with nonzero, relatively prime integer coefficients u_1,..., u_m. For any set A of integers, let F(A) = {F(a_1,...,a_m…
Dense sets of integers with prescribed representation functions
Javier Cilleruelo, Melvyn B. Nathanson
Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to t…
Inverse problems for linear forms over finite sets of integers
Melvyn B. Nathanson
Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \subseteq Z and |A|=k}. A minimizing k-set…
Sums of products of congruence classes and of arithmetic progressions
Sergei V. Konyagin, Melvyn B. Nathanson
Consider the congruence class R_m(a)={a+im:i\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m…