Optimal constants in L1 norm bounds for Fourier transforms of set indicators
Determine the optimal absolute constants in the inequalities comparing the L1 norm of the Fourier transform of 1_B to log|B| and to |B|^{1/2}, thus fixing the constant factors in the lower and upper bounds for ||\hat{1}_B||_1 for finite sets B ⊂ Z.
References
Both the upper and lower bounds are tight up to a constant factor in general.\footnote{In fact, it is a well-known problem to determine either of these constants.}
— Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series
(Bedert, 12 Feb 2025) in Section 5 (The structure of sets with small L1-norm), footnote