A Basis of Finite and Infinite Sets with Small Representation Function

  • Artūras Dubickas

Abstract

Let A be a subset of the set of nonnegative integers N{0}, and let rA(n) be the number of representations of n0 by the sum a+b with a,bA. Then (aAxa)2=n=0rA(n)xn. We show that an old result of Erdős asserting that there is a basis A of N{0}, i.e., rA(n)1 for n0, whose representation function rA(n) satisfies  rA(n)<(2e+ϵ)logn for each sufficiently large integer n. Towards a polynomial version of the Erdős-Turán conjecture we prove that for each ϵ>0 and each sufficiently large integer n there is a set A{0,1,,n} such that the square of the corresponding Newman polynomial f(x):=aAxa of degree n has all of its 2n+1 coefficients in the interval [1,(1+ϵ)(4/π)(logn)2]. Finally, it is shown that the correct order of growth for H(f2) of those reciprocal Newman polynomials f of degree n whose squares f2 have all their 2n+1 coefficients positive is n. More precisely, if the Newman polynomial f(x)=aAxa of degree n is reciprocal, i.e., A=nA, then A+A={0,1,,2n} implies that the coefficient for xn in f(x)2 is at least 2n3. 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 [1,22n+4].
Published
2012-01-06
Article Number
P6