38 research outputs found

### Limits of Boolean Functions on F_p^n

We study sequences of functions of the form F_p^n -> {0,1} for varying n, and
define a notion of convergence based on the induced distributions from
restricting the functions to a random affine subspace. Using a decomposition
theorem and a recently proven equi-distribution theorem from higher order
Fourier analysis, we prove that the limits of such convergent sequences can be
represented by certain measurable functions. We are also able to show that
every such limit object arises as the limit of some sequence of functions.
These results are in the spirit of similar results which have been developed
for limits of graph sequences. A more general, albeit substantially more
sophisticated, limit object was recently constructed by Szegedy in [Sze10].Comment: 12 page

### Measure preserving homomorphisms and independent sets in tensor graph powers

In this note, we study the behavior of independent sets of maximum
probability measure in tensor graph powers. To do this, we introduce an upper
bound using measure preserving homomorphisms. This work extends some previous
results about independence ratios of tensor graph powers.Comment: 5 page

### On the Structure of Quintic Polynomials

We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let F=F_q be a prime field. Suppose f:F^n to F is a degree five polynomial with bias(f)=delta. We prove the following two structural properties for such f.
1. We have f= sum_{i=1}^{c} G_i H_i + Q, where G_i and H_is are nonconstant polynomials satisfying deg(G_i)+deg(H_i)<= 5 and Q is a degree <5 polynomial. Moreover, c does not depend on n.
2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_V is a constant.
Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension Omega(n). Item 2.]extends this to degree five polynomials. A corollary to Item 2. is that any degree five affine disperser for dimension k is also an affine extractor for dimension O(k). We note that Item 2. cannot hold for degrees six or higher.
We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when d<|F|+4. While the d<|F|+4 assumption seems very restrictive, we note that prior to our work such structure theorems were only known for d<|F| by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n

### Structure in Communication Complexity and Constant-Cost Complexity Classes

Several theorems and conjectures in communication complexity state or
speculate that the complexity of a matrix in a given communication model is
controlled by a related analytic or algebraic matrix parameter, e.g., rank,
sign-rank, discrepancy, etc. The forward direction is typically easy as the
structural implications of small complexity often imply a bound on some matrix
parameter. The challenge lies in establishing the reverse direction, which
requires understanding the structure of Boolean matrices for which a given
matrix parameter is small or large. We will discuss several research directions
that align with this overarching theme.Comment: This is a column to be published in the complexity theory column of
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