Geodesic
A Method for Estimating the $L_1$ Norm of an Exponential Sum Based on Arithmetic Properties of the Spectrum
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 94-101
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A new lower estimate of the $L_1$ norm of a general exponential sum is established in terms of the ratios of the $L_p$ ($p>2$) and $L_2$ norms of dyadic blocks. In particular, for sums of exponents with coefficients whose absolute values are 0 and 1, the estimates are found such that the density and arithmetic properties of the spectrum are simultaneously taken into account. The results obtained are unimprovable in a certain sense. 
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     author = {S. V. Bochkarev},
     title = {A~Method for {Estimating} the $L_1$ {Norm} of an {Exponential} {Sum} {Based} on {Arithmetic} {Properties} of the {Spectrum}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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S. V. Bochkarev. A Method for Estimating the $L_1$ Norm of an Exponential Sum Based on Arithmetic Properties of the Spectrum. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 94-101. http://geodesic.mathdoc.fr/item/TM_2001_232_a9/

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