We prove two main results on how arbitrary linear threshold functions \documentclass[12pt]{minimal}
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\begin{document}$${f(x) = {\rm sign}(w \cdot x - \theta)}$$\end{document} over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is \documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document} -close to a threshold function depending only on \documentclass[12pt]{minimal}
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\begin{document}$${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$$\end{document} many variables, where \documentclass[12pt]{minimal}
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\begin{document}$${{\rm Inf}(f)}$$\end{document} denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s well-known theorem (Friedgut in Combinatorica 18(1):474–483, 1998), which states that every Boolean function f is \documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document}-close to a function depending only on \documentclass[12pt]{minimal}
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\begin{document}$${2^{O({\rm Inf}(f)/\epsilon)}}$$\end{document} many variables, for the case of threshold functions. We complement this upper bound by showing that \documentclass[12pt]{minimal}
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\begin{document}$${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$$\end{document} many variables are required for \documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document}-approximating threshold functions. Our second result is a proof that every n-variable threshold function is \documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document}-close to a threshold function with integer weights at most \documentclass[12pt]{minimal}
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\begin{document}$${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$$\end{document} This is an improvement, in the dependence on the error parameter \documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document}, on an earlier result of Servedio (Comput Complex 16(2):180–209, 2007) which gave a \documentclass[12pt]{minimal}
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\begin{document}$${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$$\end{document} bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180–209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.
