Representing integers as linear combinations of power products (Q438614)

Let \(P\) be a set of \(t\geq 2\) primes and \(A\) be the multiplicative semigroup of all positive integers whose prime factors are in \(P\). For an integer \(k\geq 2\), let \(F(k)\) be the smallest integer that cannot be represented as a sum of less than \(k\) terms from \(A\). In the paper under review, the authors give upper and lower bounds for \(F(k)\). For example, the inequalities \[ k^{ck}<F(k)<C(kt)^{(1+\varepsilon) kt} \] hold for all \(k>1\), where \(\varepsilon>0\) is arbitrary, \(c>0\) is an absolute constant and \(C\) is a positive constant depending on \(\varepsilon\). They also give an upper bound for the quantity \(F_{\pm}(k)\) defined as the smallest positive integer which cannot be represented as a sum of less than \(k\) elements in \(A\cup (-A)\). Such questions have been inspired by the recent paper [\textit{M. Nathanson}, ``Geometric group theory and arithmetic diameter'', Publ. Math. 79, No. 3-4, 563--572 (2011; Zbl 1249.11017)]. The proofs use a variety of results, like the existence of positive integers with a small Carmichael (universal exponent) function due to \textit{P. Erdős, C. Pomerance} and \textit{E. Schmutz} [``Carmichael's lambda function'', Acta Arith. 58, No. 4, 363--385 (1991; Zbl 0734.11047)], an application of the Subspace Theorem due to \textit{J.-H. Evertse} from [``On sums of \(S\)-units and linear recurrences'', Compos. Math. 53, 225--244 (1984; Zbl 0547.10008)], as well as a result of \textit{Zs. Ádám, L. Hajdu} and the reviewer [``Representing integers as linear combinations of \(S\)-units'', Acta Arith. 138, No. 2, 101--107 (2009; Zbl 1230.11044)].