On higher energy decompositions and the sum-product phenomenon
George Shakan
- 01 Nov 2019
- Vol. 167, Iss: 3, pp 599-617
TL;DR: In this paper, the authors quantitatively improved the Balog-Wooley decomposition by partitioning A ⊂ ℝ into sets B and C such that B can be partitioned into sets C and A such that
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Abstract: Let A ⊂ ℝ be finite. We quantitatively improve the Balog–Wooley decomposition, that is A can be partitioned into sets B and C such that
$
\begin{equation*}
\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}.
\end{equation*}
$
We use similar decompositions to improve upon various sum–product estimates. For instance, we show
$
\begin{equation*}
|A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.
\end{equation*}
$
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Citations
New results on sum-product type growth over fields
Brendan Murphy,Giorgis Petridis,Oliver Roche-Newton,Misha Rudnev,Ilya D. Shkredov,Ilya D. Shkredov,Ilya D. Shkredov +6 more
TL;DR: In this paper, a range of new sum-product type growth estimates over a general field is presented. But the authors do not consider the problem of estimating the growth rate of the sum product.
34
Stronger sum--product inequalities for small sets
Misha Rudnev,Ilya D. Shkredov,George Shakan +2 more
- 06 Jan 2020
TL;DR: In this article, the threshold-breaking sum-product inequality was shown to hold regardless of the characteristic of a finite subset of a field and a finite subset of the field.
30
An update on the sum-product problem
Misha Rudnev,Sophie Stevens +1 more
- 11 Oct 2021
TL;DR: In this article, the authors improved the best known sum-product estimates over the reals by showing that for a convex set, the sum product of A+A, A+AA, and A| can be approximated by A| + A|A| + |AA| under a streamlining of the arguments of Solymosi, Konyagin and Shkredov.
15
A new sum-product estimate in prime fields
TL;DR: In this paper, the authors obtained a new sum-product estimate in prime fields for sets of large cardinality using a point-plane incidence bound rather than the point-line incidence bound used by Shakan and Shkredov.
12
•Posted Content
Breaking the 6/5 threshold for sums and products modulo a prime
George Shakan,Ilya D. Shkredov +1 more
TL;DR: In this paper, the Cartesian product point-line incidence theorem and the theory of higher energies were used to prove the existence of higher energy in the case of point-lines.
11
References
On sums and products of integers
P. Erdős,Endre Szemerédi +1 more
- 01 Jan 1983
TL;DR: In this paper, the integers of the form ======¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯�ﾷ (1) were considered, where n is a sequence of integers, and n is the number of nodes in the sequence.
288
On the number of sums and products
TL;DR: In this article, the authors improved the exponent to 5/4, which is the smallest lower bound known for any n-element set A, and they showed that the lower bound can be obtained by a positive absolute constant c 1 such that g(n) ≥ n 1+c 1.
Some new results on higher energies
TL;DR: In this article, the additive energies of convex sets with small |AA| and |A(A+1) additive energies were studied and the notion of dual popular difference sets was developed.
Sums and Products from a Finite Set of Real Numbers
TL;DR: In this paper, the behavior of the function f h (k) is studied and upper and lower bounds for the minimum of |E h (A)| taken over all A with |A| = k are shown.
59
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