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

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• 作者：Art¨±ras Dubickas^a ; ^{arturas.dubickas@mif.vu.lt} ; Tomasz Schoen^b ; ^{schoen@amu.edu.pl} ; Manuel Silva^c ; ^{mnas@fct.unl.pt} ; Paulius &Scaron ; arka^d ; ^a ; ^{paulius.sarka@mif.vu.lt}
• 刊名：European Journal of Combinatorics
• 出版年：2013
• 出版时间：October, 2013
• 年：2013
• 卷：34
• 期：7
• 页码：1144-1157
• 全文大小：408 K

文摘

For two finite sets of integers and their additive energy is the number of solutions to , where and . Given finite sets with additive energy , we investigate the sizes of largest subsets and with all sums , , being different (we call such subsets co-Sidon). In particular, for we show that in the case of small energy, , one can always find two co-Sidon subsets with sizes , whenever satisfy . An example showing that this is best possible up to the logarithmic factor is presented. When the energy is large, , we show that there exist co-Sidon subsets of with sizes whenever satisfy and show that this is best possible. These results are extended (non-optimally, however) to the full range of values of .