A note on quantitative stability in Hilbert spaces

We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon )$-stable for all $k\ge \exp (\pi \mathrm{/}\epsilon )$, and it is not $(k,\epsilon )$-stable for $k\le \exp (\log 2\mathrm{/}\epsilon )$, showing that the growth is necessarily exponential in $1\mathrm{/}\epsilon$. We then analyze how stability scales under nonlinear connectives applied to the inner product. In particular, for power-type predicates $f(x,y)=⟨x,y{⟩}_{+}^{\beta }$ with $\beta <1$ we obtain upper and lower bounds of the form $\exp (C{\epsilon }^{-1\mathrm{/}\beta })$, and for $\beta >1$ and integer powers $⟨x,y{⟩}^d$ we retain the bilinear scale $\exp (C\mathrm{/}\epsilon )$.