It is proved that every n × n Latin square has a partial transversal of length at least n − O(log 2 n). The previous papers proving these results (including one by the second author) not only contained an error, but were sloppily written and quite difficult to understand. We have corrected the error and improved the clarity.

Set F = Fp for any fixed prime p2. An affine-invariant property is a property of functions over F n that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that given an input function f , we make a constant number of queries to f , always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable.We also prove that any property that can be described as the property of decomposing into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized.Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the *

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a nearequidistribution theorem that describes these distributions for the group F n p when p is a fixed prime. This fundamental fact is equivalent to a strong near-orthogonality statement regarding the higher-order characters, and was previously known only under various extra assumptions about the linear forms.As an application of our near-equidistribution theorem, we settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms for the group F n p .

We present an explicit pseudorandom generator with seed length O((log n) w +1 ) for read-once, oblivious, width w branching programs that can read their input bits in any order. This improves upon the work of Impagliazzo, Meka and Zuckerman (FOCS'12) where they required seed length n 1/2+o (1) .A central ingredient in our work is the following bound that we prove on the Fourier spectrum of branching programs. For any width w read-once, oblivious branching program B : {0, 1} n → {0, 1}, any k ∈ {1, . . . , n},This settles a conjecture posed by Reingold, Steinke and Vadhan (RANDOM'13).Our analysis crucially uses a notion of local monotonicity on the edge labeling of the branching program. We carry critical parts of our proof under the assumption of local monotonicity and show how to deduce our results for unrestricted branching programs.

We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over R) is a C ·2 d -junta, i.e. it depends on at most C ·2 d variables. This improves the d · 2 d−1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].The bound of C · 2 d is tight up to the constant C as a lower bound of 2 d − 1 is achieved by a read-once decision tree of depth d. We slightly improve the lower bound by constructing, for each positive integer d, a function of degree d with 3 · 2 d−1 − 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.