TY - JOUR
T1 - Large values of the additive energy in ℝd and ℤd
AU - Shao, Xuancheng
PY - 2014/3
Y1 - 2014/3
N2 - Combining Freiman's theorem with Balog-Szemerédi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ℝd and ℤd.
AB - Combining Freiman's theorem with Balog-Szemerédi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ℝd and ℤd.
UR - http://www.scopus.com/inward/record.url?scp=84897027757&partnerID=8YFLogxK
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U2 - 10.1017/S0305004113000741
DO - 10.1017/S0305004113000741
M3 - Article
AN - SCOPUS:84897027757
SN - 0305-0041
VL - 156
SP - 327
EP - 341
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
IS - 2
ER -