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Alex B.
Alex B.
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LowerSuch $X$ do indeed exist, and upper bounds on the number of solutions to such equations are consideredexplicitly constructed in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282, Theorem 7.5. The whole paper is devoted to upper and lower bounds on sizes of solution-free sets to equations such as the one you are asking about. A mathscinet forward search from that paper should yield further results.

Lower and upper bounds on the number of solutions to such equations are considered in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282. A mathscinet forward search should yield further results.

Such $X$ do indeed exist, and are explicitly constructed in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282, Theorem 7.5. The whole paper is devoted to upper and lower bounds on sizes of solution-free sets to equations such as the one you are asking about. A mathscinet forward search from that paper should yield further results.

Source Link
Alex B.
Alex B.
  • 13k
  • 4
  • 56
  • 90

Lower and upper bounds on the number of solutions to such equations are considered in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282. A mathscinet forward search should yield further results.