Low Soundness Linearity Testing on the Half-Slice

Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$ such that $x,y,x+y\in T$, then $f$ agrees with some linear function on at least $\frac{1+\delta}{2}-o(1)$ fraction of the points in $T$. More generally, we show that if $f$ passes the natural $k$-query BLR test with probability $\frac{1+\delta}{2}$ for any $k\geq3$, then it must agree with some affine function at $\frac{1+\delta^{\frac{1}{k-2}}}{2}-o(1)$ fraction of the points in $T$. The only other known linearity test for the slice in the low soundness regime (i.e., when $\delta$ can be arbitrarily small) was given by Kalai, Lifshitz, Minzer, and Ziegler [FOCS'24]. Our result improves upon this result in two significant ways: firstly, it works for $k=3$ queries, instead of requiring $k\geq4$; secondly, our result is sharper, e.g., when $k=4$, we are able to conclude an agreement of $\frac{1+\sqrt{\delta}}{2}-o(1)$ instead of $\frac{1+c\sqrt{\delta}}{2}$ for $c\approx.0035$. In particular, our result matches (up to the $o(1)$ term) the conclusion one obtains over the full hypercube via the classical BLR analysis. Our main technical contribution is a new dense model theorem using bounds on Krawtchouk polynomials. Using these Krawtchouk polynomial bounds, we also obtain a simple $k$-query test ($k\geq 5$) that avoids any use of the dense model machinery. This simplified test naturally extends to the slice over the $q$-ary hypercube, giving the first such result over larger alphabets.