An old conjecture of Erdos–Turán on additive bases

Abstract:

There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate: Conjecture 0.1(Erdős and Turán). Suppose that $0 = \delta _0<\delta _1<\delta _2<\delta _3\cdots$ is an increasing sequence of integers and \[ s(z) : = \sum _{i=0}^\infty z^{\delta _i}. \] Suppose that \[ s^2(z) := \sum _{i=0}^\infty b_i z^i. \] If $b_i>0$ for all $i$, then $\{b_n\}$ is unbounded. Our main purpose is to show that the sequence $\{b_n\}$ cannot be bounded by $7$. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.