Let be a large enough prime number. When is a subset of of cardinality , then an application of the Cauchy–Davenport theorem gives . In this note, we improve on this and we show that implies . In the opposite direction we show that there exists a set such that and .
"A note on the set $A(A+A)$." Mosc. J. Comb. Number Theory 8 (2) 179 - 188, 2019. https://doi.org/10.2140/moscow.2019.8.179