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We show that a Boolean degree $d$ function on the slice ${[n]} \choose k$ is a junta if $k \geq 2d$, and that this bound is sharp. We prove a similar result for $A$-valued degree $d$ functions for arbitrary finite $A$, and for functions on an infinite analog of the slice.

One of the questions left open is the restriction threshold, which is the minimal $k$ which guarantees (for large enough $n$) that an $A$-valued degree $d$ function on ${[n]} \choose k$ is the restriction of an $A$-valued degree $d$ function on $\{0,1\}^n$. The paper gives an example in which the restriction threshold is larger than the junta threshold, and conjectures that the two coincide when $A = \{0,1\}$. This short note (joint with Antoine Vinciguerra) confirms this conjecture for all $A$ which are arithmetic progressions (and in particular, for $A = \{0,1\}$).