Non-trivial solutions to a linear equation in integers
Abstract
For k>=3 let A ⊂ [1,N] be a set not containing a solution to a1 x1+...+ak xk=a1 xk+1+...+ak x2k in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A has O(N1/2-epsilon) elements. This answers a question of I. Ruzsa.
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