Multifold sums and products over R, and combinatorial problems on sumsets

Additive combinatorics;Sum-product inequalities

Bush, Albert ; Croot, Ernest Mathematics Lacey, Michael Lyall, Neil Tetali, Prasad Trotter, William ; Croot, Ernest

University:Georgia Institute of Technology

Department:Mathematics

关键词: Additive combinatorics;    Sum-product inequalities;   

Others  :  https://smartech.gatech.edu/bitstream/1853/53950/1/BUSH-DISSERTATION-2015.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We prove a new bound on a version of the sum-product problem studied by Chang.By introducing several combinatorial tools, this expands upon a method of Croot and Hart which used the Tarry-Escott problem to build distinct sums from polynomials with specific vanishing properties.We also study other aspects of the sum-product problem such as a method to prove a dual to a result of Elekes and Ruzsa and a conjecture of J. Solymosi on combinatorial geometry.Lastly, we study two combinatorial problems on sumsets over the reals.The first involves finding Freiman isomorphisms of real-valued sets that also preserve the order of the original set.The second applies results from the former in proving a new Balog-Szemeredi type theorem for real-valued sets.