On Mimicking Rademacher Sums

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For the discrete cube with fixed cutoff K the L1 distance from the Rademacher sum to the subspace of only high frequencies is Θ(K): an explicit function is constructed so its Walsh singletons equal one, all lower coefficients vanish, and its L1 norm ≤ C K, while a matching lower bound shows optimality. Construction uses Walsh expansion and a trigonometric (Fejér/triangle) polynomial to kill low modes and control L1. The paper also develops a two-to-n concentration lemma that drops one summand to force concentration, giving a new quantitative route to FKN-type stability (constants ≈5–6).

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1. 0:00Intro
2. 0:44Discrete cube - notation
3. 1:48Walsh system
4. 2:44Orthonormality
5. 5:20Question
6. 8:23Main result
7. 19:46Proof of the bound - the end
8. 29:14Proof of optimality - the end
9. 32:04On some extensions of the FKN theorem
10. 36:55Main trick
11. 41:12Reduction from n to two summands
12. 54:15Geometric problem
13. 56:47The case of n N