Dense sets of integers with prescribed representation functions

作者：Cilleruelo Javier^*; Nathanson Melvyn B

来源：European Journal of Combinatorics, 2013, 34(8): 1297-1306.

DOI：10.1016/j.ejc.2013.05.012

Let A be a set of integers and let h %26gt;= 2. For every integer n, let r(A),(h)(n) denote the number of representations of n of the form n = a(1) + ... + a(h), where a(i) is an element of A for 1 %26lt;= i %26lt;= h, and a(1) %26lt;= ... %26lt;= a(h). The function r(A,h) : Z -%26gt; N, where N = N boolean OR {0, infinity}, is the representation function of order h for A. %26lt;br%26gt;We prove that, given a positive integer g, every function f : Z -%26gt; N satisfying lim inf(vertical bar n vertical bar -%26gt;infinity) f(n) %26gt;= g is the representation function of order h of a sequence A of integers %26quot;almost%26quot; as dense as any given B-h[g] sequence. Specifically we prove that, given an integer h %26gt;= 2 and epsilon %26gt; 0, there exists g = g(h, epsilon) such that for any function f : Z -%26gt; N satisfying lim inf(vertical bar n vertical bar -%26gt;infinity)f (n) %26gt;= g there exists a sequence A satisfying r(A,h) = f and vertical bar A boolean AND [1, x] %26gt;%26gt; x((1/h)-epsilon). %26lt;br%26gt;Roughly speaking we prove that the problem of finding a dense set of integers with a prescribed representation function f of order h and lim inf vertical bar n vertical bar -%26gt;infinity f(n) %26gt;= g is %26quot;equivalent%26quot; to the classical problem of finding dense B-h [g] sequences of positive integers.

• 出版日期2013-11