On the Wiener norm of subsets of $\mathbb Z_p$ of medium size
Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 5, pp. 75-87.

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We give a lower bound for the Wiener norm of the characteristic function of a subset $A$ from $\mathbb Z_p$, where $p$ is a prime number, in the situation where $\exp((\log p/\log\log p)^{1/3})\le|A|\le p/3$.
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S. V. Konyagin; I. D. Shkredov. On the Wiener norm of subsets of $\mathbb Z_p$ of medium size. Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 5, pp. 75-87. http://geodesic.mathdoc.fr/item/FPM_2014_19_5_a3/

[1] Zigmund A., Trigonometricheskie ryady, v. 2, Mir, M., 1965 | MR

[2] Konyagin S. V., “O probleme Littlvuda”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 243–265 | MR | Zbl

[3] Konyagin S. V., Shkredov I. D., Kolichestvennyi variant teoremy Berlinga–Khelsona (to appear)

[4] Lebedev V. V., “Absolyutno skhodyaschiesya ryady Fure. Usilenie teoremy Berlinga–Khelsona”, Funkts. analiz i ego pril., 46:2 (2012), 52–65 | DOI | MR | Zbl

[5] Bourgain J., Garaev M. Z., “On a variant of sum-product estimates and explicit exponential sum bounds in prime fields”, Math. Proc. Cambridge Philos. Soc., 146:1 (2009), 1–21 | DOI | MR | Zbl

[6] Green B. J., Konyagin S. V., “On the Littlewood problem modulo a prime”, Can. J. Math., 61:1 (2009), 141–164 | DOI | MR | Zbl

[7] McGehee O. C., Pigno L., Smith B., “Hardy's inequality and the $L^1$ norm of exponential sums”, Ann. Math., 113 (1981), 613–618 | DOI | MR | Zbl

[8] Sanders T., “The Littlewood–Gowers problem”, J. Anal. Math., 101 (2007), 123–162 | DOI | MR | Zbl

[9] Sanders T., “The structure theory of set addition revisited”, Bull. Am. Math. Soc., 50:1 (2013), 93–127 | DOI | MR | Zbl

[10] Schoen T., “New bounds in Balog–Szemerédi–Gowers theorem”, Combinatorica, Accepted

[11] Tao T., Vu V., Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl