Uniform weighted averages and a conjecture of Bergelson, Moreira, and Richter

We confirm a conjecture posed by Bergelson, Moreira, and Richter, and in particular show that for every probability-measure-preserving system ( X , B , μ , T ) (X,B,μ ,T) left parenthesis upper X comma script upper B comma mu comma upper T right parenthesis , every k ∈ N k∈ N k element of double struck upper N , every set A ∈ B A∈ B upper A element of script upper B with μ ( A ) > 0 μ (A)>0 mu left parenthesis upper A right parenthesis greater than 0 , and every tempered function f , lim N → ∞ ( 1 / N ) ∑ n = 1 N μ ( A ∩ T − ⌊ f ( n ) ⌋ A ∩ T − ⌊ f ( n + 1 ) ⌋ A ∩ ⋯ ∩ T − ⌊ f <mml:mo stretchy="false