Subset sums of power values over Galois rings

Source abstract

Let $R=GR(p^m,p^{mf})$, where $p$ is prime, $q=p^f$, $m,f\geq1$, $d\geq2$ and $p\nmid d$. We count $n$-element subsets of $P=\{x^d:x\in R\}\setminus\{0\}$, the set of distinct nonzero $d$th-power values, with a prescribed sum. For fixed $m,d,n$ with $n\geq3$, the count equals $$q^{-m}\binom{|P|}{n}(1+O_{m,d,n}(q^{1-n/2}))$$ as $q\to\infty$ through admissible prime powers, uniformly over all targets and with both $p$ and $f$ allowed to vary. For fixed $m,d$, we also classify uniform relative equidistribution at the remaining positive sizes: it fails at $n=1$, and at $n=2$ it holds along an admissible sequence if and only if, for all sufficiently large terms, $\gcd(d,q-1)=1$ and $(m\geq2\text{ or }p\ne2)$.