It was shown by Davenport and Roth  that the values taken by
at integer points ( x1, …, x8) ∈ ℤ8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang  showed that for all the inequality
has infinitely many solutions in integers. Later Baker  obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable.