A representation of large integers from combinatorial sieves (Q1900884)

By using sieve arguments the author proves the following statement. Let \(k\) and \(m\) be positive integers, let \(l\) be an integer with \(0\leq l< m\). Then there are positive numbers \(\beta= \beta (k,m)\) and \(n_2= n_2 (k, m)\) such that any integer \(x\geq n_2\) can be represented as \[ x= f_1 \dots f_k+ rm+ l \] where \(f_1, \dots, f_k\), and \(r\) are nonnegative integers with \(rm+l\leq x^\beta\) and \(f_i\geq x^\beta\) \((i=1, \dots, k)\). The author announces applications to the construction of orthogonal arrays and MDS codes (to appear).