On the Sensitivity of Cyclically-Invariant Boolean Functions
<|reference_start|>On the Sensitivity of Cyclically-Invariant Boolean Functions: In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta(n^{1/3})$. This result answers two previously published questions. Tur\'an (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity $\Omega(\sqrt{n})$. Kenyon and Kutin (2004) asked whether for a ``nice'' function the product of 0-sensitivity and 1-sensitivity is $\Omega(n)$. Our function answers both questions in the negative.   We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is $\Omega(n^{1/3})$. Hence for this class of functions sensitivity and block sensitivity are polynomially related.<|reference_end|>
@article{chakraborty2005on,
    title={On the Sensitivity of Cyclically-Invariant Boolean Functions},
    author={Sourav Chakraborty},
    journal={arXiv preprint arXiv:cs/0501026},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0501026},
    primaryClass={cs.CC}
}