117 research outputs found
Solution-free sets for linear equations
AbstractWe study the maximum size and the structure of sets of natural numbers which contain no solution of one or more linear equations. Thus, for every natural i and k⩾2, we find the minimum α=α(i,k) such that if the upper density of a strongly k-sum-free set A⊆N is at least α, then A is contained in a maximal strongly k-sum-free set which is a union of at most i arithmetic progressions. We also determine the maximum density of sets of natural numbers without solutions to the equation x=y+az, where a is a fixed integer
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