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Given a linear equation of the forma  1    x  1   +   a  2    x  2   +   a  3    x  3   =  0    with integer coefficients    a  i   , we are interested in maximising the number of solutions to this equation in a set    S  ⊆  Z   , for sets   S   of a given size.  We prove that, for any choice of constantsa  1   ,   a  2and    a  3   , the maximum number of solutions is at least(   1  12   +  o   (  1  )   )|  S  |   2    . Furthermore, we show that this is optimal, in the following sense. For any    ε  >  0   , there are choices ofa  1   ,   a  2and    a  3   , for which any large set   S   of integers has at most(   1  12   +  ε  )|  S  |   2     solutions.  For equations in    k  ⩾  3    variables, we also show an analogous result. Setσ  k   =   ∫   −  ∞   ∞(    sin  π  x    π  x    )   k   d  x   . Then, for any choice of constantsa  1   ,  …  ,   a  k    , there are sets   S   with at least(    σ  k    k   k  −  1+  o   (  1  )   )|  S  |    k  −  1solutions toa  1    x  1   +  ⋯  +   a  k    x  k   =  0   . Moreover, there are choices of coefficientsa  1   ,  …  ,   a  kfor which any large set   S   must have no more than(    σ  k    k   k  −  1+  ε  )|  S  |    k  −  1solutions, for any    ε  >  0   .

Maximising the number of solutions to a linear equation in a set of integers