Research proof index

Let G be a finite group acting on S, and R be the fixed subring. If Q is a prime of S with finite residue field, then there exists a Frobenius element σ : G at Q.

If F has a Fourier expansion ∑_{m≥0} a_m exp(2πi(m+n₀)z) with n₀ > 0, and the coefficients are absolutely summable at height im z = c, then F = O(exp(-2π n₀ · im z)) at atImInfty. The key bound is: for im z ≥ c, ‖F(z)‖ ≤ (∑_m ‖a_m‖ · exp(-2π c m)) · exp(-2π n₀ · im z)

If F : ℍ → ℂ is O(exp(-c * im τ)) at atImInfty for some c > 0, then the restriction to the imaginary axis t ↦ F(it) is O(exp(-c * t)) at atTop.

The set of large left jump times has no accumulation points. TODO: maybe to_dual can be extended to simplify this proof as the proof of the second part is very similar to the first part.

Suppose the compatible phase system satisfies the following oscillatory cancellation estimate on every ball $B(x,r)$: for every Lipschitz amplitude $\varphi$ supported in the ball and every pair of phases $f,g$,

$\left\lVert\int_{B(x,r)}e^{i(f-g)}\varphi\right\rVert\le A\,\mu(B(x,r))\,\lVert\varphi\rVert_{\mathrm{Lip}}\,(1+d_{x,r}(f,g))^{-\tau}.$

Then the metric phase space satisfies the project’s IsCancellative property with exponent $\tau$.

Transport a compact system along an equivalence of types.