On the density of bounded bases

Abstract

Abstract For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $a\leqslant a'$ and $a, a'\in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'\in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=\sum_{i\in S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=\emptyset$. For real number x, let $A(-x,x)$ be the number of elements $a\in A$ with $-x\leqslant a\leqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)\geqslant 1$ for all sufficiently large n, then $\limsup_{n\rightarrow\infty}r_A(n)=\infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2\log x/\log 5+c_1\leqslant A(-x,x)\leqslant 2\log x/\log 3+c_2$ for all $x\geqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$ and $A(-x,x) \gt (4/\log 5)\log\log x+c$ for all $x\geqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.