## Finding large co-Sidon subsets in sets with a given additive energy

### Authors:

- Arturas Dubickas,
- Tomasz Schoen,
- Manuel Silva,
- Paulius Šarka

### Abstract

For two finite sets of integers A and B their additive energy E (A, B) is the number of solutions to a + b = ^{a '} + ^{b '}, where a, ^{a '} ∈ A and b, ^{b '} ∈ B. Given finite sets A,B⊆Z with additive energy E (A, B) = | A | | B | + E, we investigate the sizes of largest subsets ^{A '} ⊆ A and ^{B '} ⊆ B with all |^{A '} | | ^{B '} | sums a + b, a ∈ ^{A '}, b ∈ ^{B '}, being different (we call such subsets ^{A '}, ^{B '} co-Sidon). In particular, for |A | = | B | = n we show that in the case of small energy, n ≤ E = E (A, B) - | A | | B | ≪ ^{n2}, one can always find two co-Sidon subsets ^{A '}, ^{B '} with sizes |^{A '} | = k, | ^{B '} | = ℓ, whenever k, ℓ satisfy k^{ℓ2} ≪ ^{n4} / E. An example showing that this is best possible up to the logarithmic factor is presented. When the energy is large, E ≫ ^{n3}, we show that there exist co-Sidon subsets ^{A '}, ^{B '} of A, B with sizes |^{A '} | = k, | ^{B '} | = ℓ whenever k, ℓ satisfy kℓ ≪ n and show that this is best possible. These results are extended (non-optimally, however) to the full range of values of E. © 2013 Elsevier Ltd.

- Record ID
- UAMa1358bd6265a47aea8dab0ea87d82d81
- Author
- Journal series
- European Journal of Combinatorics, ISSN 0195-6698
- Issue year
- 2013
- Vol
- 34
- Pages
- 1144-1157
- ASJC Classification
- DOI
- DOI:10.1016/j.ejc.2013.04.002 opening in a new tab
- Language
- en English
- Score (nominal)
- 30
- Score source
- journalList
- Score
- Publication indicators
- = 0.000; : 2013 = 1.423; : 2013 = 0.612 (2) - 2013=0.732 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMa1358bd6265a47aea8dab0ea87d82d81/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.

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