Integer sequences and semidefinite programming
A theorem of \textit{K. F. Roth} [Acta Arith. 9, 257-260 (1964; Zbl 0125.29601)] asserts that for any sequence \(x_1, \dots, x_n\) of \(\pm 1\)'s there is an arithmetical progression \(A\) such that \(|\sum _{i\in A} x_i|\gg n^{1/4} \). The proof is simple and beautiful. The author considers a related square mean, which is a linear function in the variables \(y_{ij}=x_ix_j\). Then the restriction that \(y_{ij}\) is of this form is replaced by the weaker assumption that the matrix \((y_{ij})\) is positive semidefinite, and the resulting optimization problem is solved.
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