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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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