On Almost k-Covers of Hypercubes

In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in ℝn, such that all the vertices of 0→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow 0$$\end{document} are covered at least k times, and {0,1}n\{0→}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\{{0,1} \right\}^n}\backslash \left\{{\overrightarrow 0} \right\}$$\end{document} is uncovered? The k = 1 case is the well-known Alon-Füredi theorem which says a minimum of n affine hyperplanes is required, which follows from the Combinatorial Nullstellensatz. We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integral version for fixed n and k → ∞. We also use a Punctured Combinatorial Nullstellensatz developed by Ball and Serra, to show that a minimum of n + 3 affine hyperplanes is needed for k = 3, and pose a conjecture for arbitrary k and large n.