Geodesic
A~Method for Estimating the $L_1$ Norm of an Exponential Sum Based on Arithmetic Properties of the Spectrum
Informatics and Automation, 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. 
@article{TRSPY_2001_232_a9,
     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 = {Informatics and Automation},
     pages = {94--101},
     publisher = {mathdoc},
     volume = {232},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a9/}
}
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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. Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 94-101. http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a9/