A sequences analog of the Piatetski-Shapiro problem
This paper studies the occurrence of numbers of the form \([n^c]\) in sumsets and product sets. Theorem 1 asserts that for any \(1<c<13/11\) there is a \(\delta >0\) such that for every pair of sets \(A, B \subset [N/2, N ]\) satisfying \( ( | A | | B |) ^{1/2} \gg N^{1-\delta } \), \(\delta =\delta (c)\) there are numbers \(a\in A, b\in B\) such that \(ab=[n^c]\). For the corresponding infinite problem the range is extended to \(1<c<6/5\). Theorem 2 shows that this cannot be extended for any \(c>2\); in fact, the proof shows that \([n^c]\) cannot be replaced by any sequence \( (h_n)\) satisfying \(h_n \gg n^c\), \(c>2\), though this is not stated. In Theorem 3 the solvability of \(a+b=[n^c]\) is established for any nonintegral \(c\), under similar cardinality assumptions on \(A\) and \(B\). Curiously the result is stated only for \(c>12\) (albeit a result for a fixed \(c\) immediately implies a similar result for any \(c/k\) with integer \(k\)). The proofs use exponential sums, in the additive case in a more or less standard way, in the multiplicative case in special variants as developed recently by several people, to a considerable extent by the authors.
scientific article; zbMATH DE number 1336389