Additivity of the Gerlits--Nagy property and concentrated sets

We settle all problems posed by Scheepers, in his tribute paper to Gerlits, concerning the additivity of the Gerlits–Nagy property and related additivity numbers. We apply these results to compute the minimal number of concentrated sets of reals (in the sense of Besicovitch) whose union, when multiplied with a Gerlits–Nagy space, need not have Rothberger’s property. We apply these methods to construct a large family of spaces, whose product with every Hurewicz space has Menger’s property.