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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 28, 2016 17:00–17:25, Dorodnitsyn Computing Centre, Department of Mechanics and Mathematics of Lomonosov Moscow State University., 119991, Moscow, Gubkina str., 8, Steklov Mathematical Institute, 9 floor, Conference hall
 


Characters sums with additive convolutions

I. D. Shkredovabc

a Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
c Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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I. D. Shkredov


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Abstract: Let $\chi(x)$ be a nontrivial multiplicative character over prime modulo $p$, and $A$, $B$ be arbitrary subsets of $\mathbb{Z}/p\mathbb{Z}$ such that $|A+A| \le K|A|$, where $K \ge 1$ be a constant and $|A|,|B|> p^{ 4/9+\varepsilon}$, $\varepsilon>0$.

M.-C. Chang obtained a nontrivial upper bound for the sum
$$ |\sum_{a\in A,  b\in B} \chi(a+b)| \ll_{K,\varepsilon} |A||B|\cdot p^{-\tau(K,\varepsilon)}, \qquad (1) $$
where $\tau(K,\varepsilon)>0$.

Recently, B. Hanson considered an analog of sum (1) for three sets $A$, $B$ $C$ having no restrictions on its sumsets. Namely, he proved that if $|A|,|B|,|C| > \delta \sqrt{p}$, where $\delta>0$, then
$$ |\sum_{a\in A,  b\in B,  c\in C} \chi(a+b+c)| =  o_{\delta}(|A||B||C|). \qquad (2) $$


Using the almost periodicity lemma of Croot–Sisask as well as new results on sum-products, we refine both (1) and (2).

Language: Russian and English

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