Spectral factorization of trigonometric polynomials and lattice geometry (Q2919693)

It was shown by \textit{F. Riesz} [J. Reine Angew. Math. 146, 83--87 (1915; JFM 45.0408.01)] that if \(f(X)\) is a non-negative trigonometric polynomial having frequencies in the difference set \(F-F\), where \(F\) is an interval \([0,n]\), then \(f(X)=|g(X)|^2\), where \(g(X)\) is a trigonometrical polynomial with frequencies in \(F\). The author conjectures the following analogue of this result: if \(\alpha,\beta\) are real, and \(F\) is the set of lattice points \((x,y)\) in the region bounded by the lines \(\alpha x+\beta,\alpha x-\beta\), and \(f(X,Y)\) is a non-negative trigonometric polynomial having frequencies in \(F-F\), then \(f\) is a uniform limit of a sequence \(|f_n(X,Y)|^2\), where \(f_n\) are trigonometric polynomials with frequencies in \(F\). He proves this conjecture in the cases when \(\alpha\) is either rational or there exists a sequence of matrices \(g^{(n)}\in SL_2(Z)\) such that NEWLINE\[NEWLINE\lim_{n\to\infty}|g{(n)}_{21}+\alpha g{(n)}_{22}||g{(n)}_{22}|\log(|g{(n)}_{22}|)=0.NEWLINE\]