The first lower bound was proved by Jayram and Woodruff [1] and then by Kane,Meka,Nelson [2]. The lower bound tells that any ( , δ)-DJL for , δ ∈ (0, 1/2) must have m = Ω(min{n, −2 log(1δ )}). The second proof builds on the following idea: Since for all x we have the probabilistic guarantee PΠ∼D ,δ [|‖Πx‖2−1| < max{ , 2}] < δ, then it is true also for any distribution over x. We are going to pi...