Low-energy decomposition results over finite fields

We prove various low-energy decomposition results, showing that we can decompose a finite set $A\subset \mathbb{F}_p$ satisfying $|A|<p^{5/8}$, into $A = S\sqcup T$ so that, for a non-degenerate quadratic $f\in \mathbb{F}_p[x,y]$, we have \[ |\{(s_1,s_2,s_3,s_4)\in S^4 : s_1 + s_2 = s_3 + s_4\}| \ll |A|^{3 - \frac15 + \varepsilon} \] and \[ |\{(t_1,t_2,t_3,t_4)\in T^4 : f(t_1, t_2) = f(t_3, t_4)\}|\ll |A|^{3 - \frac15 + \varepsilon}\,. \] Variations include extending this result to large $A$ and a low-energy decomposition involving additive energy of images of rational functions. This gives a quantitative improvement to a result of Roche-Newton, Shparlinski and Winterhof as well as a generalisation of a result of Rudnev, Shkredov and Stevens. We consider applications to conditional expanders, exponential sum estimates and the finite field Littlewood problem. In particular, we improve results of Mirzaei, Swaenepoel and Winterhof and Garcia.