A basis of finite and infinite sets with small representation function

作者：Dubickas Arturas^*

来源：Electronic Journal of Combinatorics, 2012, 19(1): P6.

Let A be a subset of the set of nonnegative integers N boolean OR {0}, and let r(A)(n) be the number of representations of n %26gt;= 0 by the sum a + b with a, b is an element of A. Then (Sigma(a is an element of A) x(a))(2) = Sigma(infinity)(n=0) r(A)(n)x(n). We show that an old result of Erdos asserting that there is a basis A of N boolean OR{0}, i.e., r(A)(n) %26gt;= 1 for n %26gt;= 0, whose representation function r(A)(n) satisfies r(A)(n) %26lt; (2e + epsilon) log n for each sufficiently large integer n. Towards a polynomial version of the Erdos-Turan conjecture we prove that for each epsilon %26gt; 0 and each sufficiently large integer n there is a set A subset of {0, 1,...,n} such that the square of the corresponding Newman polynomial f (x) := Sigma(xa)(a is an element of A) of degree n has all of its 2n + 1 coefficients in the interval [1, (1 + epsilon)(4/pi)(log n)(2)]. Finally, it is shown that the correct order of growth for H(f(2)) of those reciprocal Newman polynomials f of degree n whose squares f(2) have all their 2n + 1 coefficients positive is root n. More precisely, if the Newman polynomial f(x) = Sigma(xa)(a is an element of A) of degree n is reciprocal, i.e., A = n - A, then A + A = {0, 1,...,2n} implies that the coefficient for x(n) in f(x)(2) is at least 2 root n - 3. In the opposite direction, we explicitly construct a reciprocal Newman polynomial f (x) of degree n such that the coefficients of its square f (x)(2) all belong to the interval [1, 2 root 2n + 4]

• 出版日期2012-1-6