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

Improved bounds on the set A(A+1)