Sharp inequality |A1+⋯+An| ≥ (∏|Ai|)^{1/p} holds with p = n log(m+1)/log(nm+1) for Ai ⊆ {0..m}^d, exponent optimal, obtained from a functional inequality on Z^d.
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Introduces Beckmann boundary B(f) as inf E||V||_2 over div V = Lf and proves B(f) ≳ Var(f) sqrt(log(1 + 1/sum Inf_i(f)^2)) for nonconstant Boolean f, with sharp one-sided fractional spectral bounds.
Sharp lower bounds ||S_p(1_A)||_1 ≳ |A|* log(1/|A|*) are established for dyadic square functions S1 and S2 on indicators, using Brownian exit times and the Takagi function.
Dictator functions maximize Φ-stability locally for balanced Boolean functions; computer methods confirm Courtade-Kumar conjecture for ρ≤0.914 and symmetrized Li-Médard for q∈[1.36,2).
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Sharp Lower Bounds for Sumsets in Hypercubes
Sharp inequality |A1+⋯+An| ≥ (∏|Ai|)^{1/p} holds with p = n log(m+1)/log(nm+1) for Ai ⊆ {0..m}^d, exponent optimal, obtained from a functional inequality on Z^d.
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A Beckmann boundary form of Talagrand's conjecture on the discrete cube
Introduces Beckmann boundary B(f) as inf E||V||_2 over div V = Lf and proves B(f) ≳ Var(f) sqrt(log(1 + 1/sum Inf_i(f)^2)) for nonconstant Boolean f, with sharp one-sided fractional spectral bounds.
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Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets
Sharp lower bounds ||S_p(1_A)||_1 ≳ |A|* log(1/|A|*) are established for dyadic square functions S1 and S2 on indicators, using Brownian exit times and the Takagi function.
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Local Optimality of Dictator Functions with Applications to Courtade--Kumar and Li--M\'edard Conjectures
Dictator functions maximize Φ-stability locally for balanced Boolean functions; computer methods confirm Courtade-Kumar conjecture for ρ≤0.914 and symmetrized Li-Médard for q∈[1.36,2).