Thin bases of order \(h\). (Q1869780)

The author calls a set \(A\) of nonnegative integers a unique-representation (u.r.) basis of order \(h\) if it has a decomposition \(A = \bigcup _{j=1}^h A_j\) such that every nonnegative integer has a unique representation of the form \(a_1+\dots +a_h\) with \(a_j\in A_j\). For every \(h\) and every \(\varepsilon >0\) a u.r. basis is constructed which satisfies \( \limsup A(n)n^{-1/h} \leq \beta _h + \varepsilon \), where \(\beta _h = (2^h-1)^{1/h} (2^{1/h}-1)^{-1} 8^{-(h-1)/(2h)}\). The author conjectures that this estimate is optimal and confirms it in the case \(h=2\).