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Solving a linear equation in a set of integers. I. (English) Zbl 1042.11525
Let integer coefficients \(a_1,\cdots,a_k\) and \(b\) be given, and consider integer sets \(\mathcal A\in[1,N]\) for which the equation \(a_1x_1+\cdots+a_kx_k=b\) has no solutions with \(x_i\in\mathcal A\). One defines \(r(N)\) as the maximal size of sets \(A\) with no “nontrivial” solution, and \(R(N)\) as the maximal size for sets with no solution in distinct integers. Well-known examples are the equations \(x_1-2x_2+x_3=0\) (“no three in arithmetic progression”) and \(x_1+x_2-x_3-x_4=0\) (Sidon sets). A large number of upper and lower estimates for \(r(N)\) and \(R(N)\) are given, depending on the structure of the equation in question. To give just one example, it is shown that for “symmetric” equations \(a_1x_1+\cdots+a_kx_k=a_1x_{k+1}+\cdots +a_kx_{2k}\), for any \(k\geq 3\) and any \(\epsilon>0\) there are coefficients \(a_1,\cdots,a_k\) such that \(R(N)\gg N^{1/2-\epsilon}\).

MSC:
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11B75 Other combinatorial number theory
11D04 Linear Diophantine equations
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