Research proof index

A dependent-random-choice corollary. Let $A$ be nonempty, let $0<\varepsilon\le 1$ and $\delta>0$, and let $p$ be a nonzero even integer satisfying $\varepsilon^{-1}\log(2/\delta)\le p$. Then there are sets $A_1,A_2$ such that the normalized difference distribution $\mu_{A_1}\mathbin{\circleddash}\mu_{A_2}$ assigns mass at least $1-\delta$ to the source's sifted set $s_{p,\varepsilon}(A)$. Both sets retain explicit density: $\operatorname{dens}(A_i)\ge \tfrac14\operatorname{dens}(A)^{2p}$ for $i=1,2$.

Generic derivative bound for σ_k q-series on compact sets. Uses σ_k(n) ≤ n^(k+1) (sigma_bound) and iter_deriv_comp_bound3 for exponential decay.

Generic summability for n^a * σ_b(n) * exp(2πinz) series. Uses σ_b(n) ≤ n^(b+1) (sigma_bound) and a33 (a+b+1) for exponential summability.

If G is generated by a set s, then the slash action by elements in G is uniquely determined by the slash action by elements in s. See slashaction_generators' for a version where s is a set of elements in G.

If G is generated by a set s, then the slash action by elements in G is uniquely determined by the slash action by elements in s. See slashaction_generators for a version where s is a set of elements in SL(2, ℤ).

This gives an upper bound on the number of points in the sphere packing X with norm less than R.