Non-trivial solutions to a linear equation in integers

For k ≥ 3 let A ⊂ [1, N ] be a set not containing a solution to a1x1 + · · · + akxk = a1xk+1 + · · · + akx2k in distinct integers. We prove that there is an ε > 0 depending on the coefficients of the equation such that every such A has O(N) elements. This answers a question of I. Ruzsa. Introduction The problems of estimating the size of a largest subset of [1, N ] not containing a solution to a given linear equation arise very frequently in combinatorial number theory. For example, the sets with no non-trivial solutions to x1 + x2 − 2x3 = 0 and x1 + x2 = x3 + x4 are sets with no arithmetic progressions of length three, and Sidon sets respectively. For several of the more prominent equations, like the two equations above, there are large bodies of results that deal with the structure and the size of solution-free sets. The first systematic study of general linear equations was undertaken by Ruzsa [Ruz93, Ruz95]. To ascribe a precise meaning to the concept of a “set with no non-trivial solution” he introduced two definitions of a trivial solution. One of them is that a solution is non-trivial if all the variables are assigned different values. For a fixed linear equation denote by R(N) the size of a largest set of integers in [1, N ] with no solution to the equation in distinct integers. Ruzsa [Ruz93] showed that if k ≥ 2, then for the symmetric equation a1x1 + . . . + akxk = a1xk+1 + · · ·+ akx2k (1) one has R(N) = O(N1/2). For k = 2, the estimate is tight for the Sidon equation. Ruzsa gave the example of equation x1 + d(x2 + . . . + xk) = xk+1 + d(xk+2 + . . . + x2k) (2) and the set A of integers in [1, N ] whose development in the base d2k consists only of digits 0, . . . , d − 1. Since addition of elements of A in (2) involves no carries in base d2k, for every solution to (2) with elements of A one necessarily has that x1 and xk+1 are equal digit by digit. Thus x1 = xk+1 implying that A contains no solution (2) in distinct integers. Since |A| = Ω(N1/2−c/ log d), this example shows that there is no ε > 0 such that the estimate R(N) = O(N1/2−ε) holds for all equations of the