claimroutineinferred_dep

Modulated packets transport along tubes

Shows modulated packets travel along tubes with amplitude ≥1/2\ge1/2 on cores
gives pointwise output size of each packet
Claim·§C Disjoint packets and the weak spacetime norm·inside propWeak-type endpoint smoothing fails logarithmically

Statement (paper)

Let λ=3N/2\lambda=3N/2, fν(x)=eiλvν⋅xbν(x)f_\nu(x)=e^{i\lambda v_\nu\cdot x}b_\nu(x) and QN=β(∣D∣/N)Q_N=\beta(|D|/N). Then, uniformly in ν,x\nu,x and 1≤t≤21\le t\le2, U(t)QNfν(x)=eiλ(vν⋅x−t) bν(x−tvν)+On(N−1w−2)U(t)Q_N f_\nu(x)=e^{i\lambda(v_\nu\cdot x-t)}\,b_\nu(x-tv_\nu)+O_n(N^{-1}w^{-2}), and this error tends to zero since w≍nδw\asymp_n\delta and N≥δ−4N\ge\delta^{-4}. Moreover, with I0=[3/2,7/4]I_0=[3/2,7/4], bν(x−tvν)=1b_\nu(x-tv_\nu)=1 on Sν×I0S_\nu\times I_0, so ∣U(t)QNfν∣≥1/2|U(t)Q_N f_\nu|\ge1/2 on Sν×I0S_\nu\times I_0 for large NN.

In plain English

For the modulated packet fν(x)=eiλvν⋅xbν(x)f_\nu(x)=e^{i\lambda v_\nu\cdot x}b_\nu(x) with λ=3N/2\lambda=3N/2 and QN=β(∣D∣/N)Q_N=\beta(|D|/N), the evolution U(t)QNfνU(t)Q_Nf_\nu equals eiλ(vν⋅x−t)bν(x−tvν)e^{i\lambda(v_\nu\cdot x-t)}b_\nu(x-tv_\nu) up to an error On(N−1w−2)O_n(N^{-1}w^{-2}) that tends to zero for 1≤t≤21\le t\le2. Consequently ∣U(t)QNfν∣≥1/2|U(t)Q_Nf_\nu|\ge1/2 on Sν×I0S_\nu\times I_0 with I0=[3/2,7/4]I_0=[3/2,7/4] for large NN.

Why it's here

This shows each frequency-localized packet is carried rigidly along its tube at unit speed with only a small error, so its output core SνS_\nu really carries amplitude of size about 1. Combined with the tube arrangement, where the cores SνS_\nu are compressed into the small set EE, it gives the pointwise lower bound on EE that the random-sign argument in the next claim uses to bound the weak norm from below.
Classified routine:
Standard wave-packet approximation: a high-frequency modulated bump evolves under the half-wave propagator by rigid translation along the direction of propagation, with the error controlled by a stationary-phase / Taylor expansion of the symbol |ξ| about λv_ν (as in Wolff's and Bourgain's Kakeya-type counterexamples).

Uses

claimTube arrangement: disjoint start, compressed end

Used by

claimRandom signs give weak-norm lower bound

Sources

PaperOpen
Section
§C Disjoint packets and the weak spacetime norm
Page
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sections/weak.tex:62
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Modulated packets transport along tubes" rests on claim-prop-weak-obstruction-1