Improved bounds on the set

For a subset A of a field F, write A ( A + 1 ) for the set { a ( b + 1 ) :   a , b ∈ A } . We establish new estimates on the size of A ( A + 1 ) in the case where F is either a finite field of prime order, or the real line. finite field case we show that A ( A + 1 ) is of cardinality at least | A | 57 / 56 − o ( 1 ) , so long as | A | < p 1 / 2 . In the real case we show that the cardinality is at least | A | 24 / 19 − o ( 1 ) . These improve on the previously best-known exponents of 106 / 105 − o ( 1 ) and 5/4 respectively.