Random Diophantine inequalities of additive type
  Using the Davenport-Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree $k$ in more than $2k$ variables the expected asymptotic formula for the density of solutions holds true. This appears to be the first metric result on Diophantine inequalities. 
J\"org Br\"udern, Rainer Dietmann
math.NT