Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor ,-\lfloor n^c\rfloor )$

For every $c\in (1,23\mathrm{/}22)$ and every probability dynamical system $(X,\mathcal{B},\mu ,T)$ we prove that for any $f,g\in L_{\mu }^{\mathrm{\infty }}(X)$ the bilinear ergodic averages $$\frac{1}{N}\sum _{n=1}^{N}f(T^{\lfloor n^c\rfloor }x)g(T^{-\lfloor n^c\rfloor }x)\text{converge for }\mu \text{-a.e. }x\in X\text{.}$$ In fact, we consider more general sparse orbits $(\lfloor h(n)\rfloor ,-\lfloor h(n)\rfloor {)}_{n\in \mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions. This is the first pointwise result for bilinear ergodic averages taken along deterministic sparse orbits where modulation invariance is present.