A Basis of Finite and Infinite Sets with Small Representation Function

Abstract

Let $A$ be a subset of the set of nonnegative integers $\mathbb{N}\cup\{0\}$, and let $r_A(n)$ be the number of representations of $n\geq 0$ by the sum $a+b$ with $a,b \in A$. Then $\big(\sum_{a \in A}x^a\big)^2=\sum_{n=0}^{\infty} r_A(n)x^n$. We show that an old result of Erdős asserting that there is a basis $A$ of $\mathbb{N}\cup \{0\}$, i.e., $r_A(n) \geq 1$ for $n \geq 0$, whose representation function $r_A(n)$ satisfies  $r_A(n) < (2e+\epsilon)\log n$ for each sufficiently large integer $n$. Towards a polynomial version of the Erdős-Turán conjecture we prove that for each $\epsilon>0$ and each sufficiently large integer $n$ there is a set $A \subseteq \{0,1,\dots,n\}$ such that the square of the corresponding Newman polynomial $f(x):=\sum_{a \in A} x^a$ 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 $\sqrt{n}$. More precisely, if the Newman polynomial $f(x)=\sum_{a \in A} x^a$ of degree $n$ is reciprocal, i.e., $A=n-A$, then $A+A=\{0,1,\dots,2n\}$ implies that the coefficient for $x^n$ in $f(x)^2$ is at least $2\sqrt{n}-3$. 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 $[1, 2\sqrt{2n}+4]$.