On the distribution of \(B_ 3\)-sequences (Q1914033)

Let \(h\in \mathbb{N}\). \(A\subseteq \mathbb{N}\), \(A\neq \emptyset\), \(|A|= \infty\) is called a \(B_h\)-sequence if every \(n\in \mathbb{N}\) has at most one representation \(n= a_1+ \dots+ a_h\) with \(a_i\in A\), \(a_1\leq \dots\leq a_h\). An old conjecture of P. Erdös states that the counting function \(A(n):= |A\cap [1, n]|\) of \(A\) satisfies \(\liminf_{n\to \infty} A(n) n^{-1/h} =0\). This has been proved for all even \(h\) by the author of this paper [Acta Arith. 63, No. 4, 367-371 (1993; Zbl 0770.11010)] and by others independently, but for odd \(h\) the problem is still open even in the case \(h=3\). Here it is proved that an infinite set \(A\subset \mathbb{N}\) satisfying \(A(n) \sim cn^{1/3}\) for some \(c>0\) cannot be a \(B_3\)-sequence. From this theorem it follows immediately that a \(B_3\)-sequence \(A\) with \(\liminf_{n\to \infty} A(n) n^{1/3}> 0\) must oscillate between intervals with a high density of elements and such with a low density. More information is given by the following theorem: Assume that \(A\) is a \(B_3\)-sequence with \(\liminf_{n\to \infty} A(n) n^{1/3} >0\). Then for every \(\delta >0\), \(\alpha\in (0, 1/2]\) and sufficiently large \(N\in \mathbb{N}\) there exists a natural number \(\ell_0\), \(\ell_0< N^\alpha\) such that \[ A(\ell N)- A((\ell- 1)N)< {{N^{1/3}} \over {\ell^{2/3} \log^{1/6- \delta} N}} \] for all \(\ell\in \mathbb{N}\) with \(\ell_0< \ell\leq \ell_0+ [\log^{1/6} N]\).