[3] Najer won the Cappelle-la-Grande Open of 2004 on tiebreak over Kaido Külaots, Artyom Timofeev, Zoltan Gyimesi, Sergey Grigoriants and Oleg Korneev.

[4] In the same year he tied for 1st–3rd with Michael Roiz and Leonid Gofshtein in the Ashdod Chess Festival.

[9] In July 2009, Najer won the strong rapid round-robin tournament, whose field included Boris Gelfand and Judit Polgar among others, of the Richard Riordan Chess Festival[10] at the 2009 Maccabiah Games.

Najer won the 2016 Aeroflot Open edging out Boris Gelfand on tiebreak, after both scored 6½/9 points; this achievement earned him a spot in the 2016 Dortmund Sparkassen Chess Meeting.

[15] In 2017, he tied for first place with Emil Sutovsky in the 18th Karpov Chess Tournament in Poikovsky, Russia.