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We study additive equations of the form∑   i  =  1   s    λ  i   P   (   n  i   )   =  0    in variablesn  i   ∈   Z  d    , where the    λ  i    are nonzero integers summing up to zero and    P  =  (   P  1   ,  ⋯  ,   P  r   )    is a system of homogeneous polynomials making the equation translation‐invariant. We investigate the solvability of this equation in subsets of density(  log  N  )    −  c  (  P  ,   λ   )of a large box[  N  ]   d    via the energy‐increment method. We obtain positive results for roughly the number of variables currently needed to derive a count of solutions in the complete box[  N  ]   d    for the multidimensional systems of large degree studied by Parsell, Prendiville and Wooley. Appealing to estimates from the decoupling theory of Bourgain, Demeter and Guth, we also treat the cases of the monomial curve    P  =  (  x  ,  ⋯  ,   x  k   )    and the parabola    P  =  (  x  ,  |  x   |  2   )    for a number of variables close to or equal to the limit of the circle method.

Additive equations in dense variables via truncated restriction estimates