The question here about estimating positive moments of the divisor function on sets of nonzero density $A\subset \{1,2,\ldots,x\}$ was answered giving $$ S_a(x):=\sum_{n \in A} d(n)^a \geq |A|(\ln x)^{a(\ln 2-\delta)}\quad (1) $$ for any $\delta>0,$ arbitrarily small.
This relies on the fact that after removing rare large values the divisor function in $\{1,2,\ldots x\}$ is essentially of size $(\ln x)^{\ln 2}$ on average.
What if I now "clip" the divisor function at level $m=(\ln n)^{2 \ln 2}$? So consider lower bounding the sum $$ S_{a,m}(x):=\sum_{n \in A} \left\{\max(d(n),m)\right\}^a $$
It seems to me that this should make no difference to the lower bound in (1), but I am not sure how to make this rigorous.
