Craig V. Spencer

On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions

We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier transform matrix. Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.

Improved bounds for a deterministic sublinear-time Sparse Fourier Algorithm

This paper improves on the best-known runtime and measurement bounds for a recently proposed Deterministic sublinear-time Sparse Fourier Transform algorithm (hereafter called DSFT). In (Iwen, 2008 ), (Iwen, 2007), it is shown that DSFT can exactly reconstruct the Fourier transform (FT) of an N-bandwidth signal f, consisting of B Lt N non-zero frequencies, using O(B2ldrpolylog(N)) time and O(B2 ldr polylog(N)) f-samples. DSFT works by taking advantage of natural aliasing phenomena to hash a frequency- sparse signals FT information modulo O(B ldr polylog(N)) pairwise coprime numbers via O(B ldr polylog(N)) small Discrete Fourier Transforms. Number theoretic arguments then guarantee the original DFT frequencies/coefficients can be recovered via the Chinese Remainder Theorem. DSFTs usage of primes makes its runtime and signal sample requirements highly dependent on the sizes of sums and products of small primes. Our new bounds utilize analytic number theoretic techniques to generate improved (asymptotic) bounds for DSFT. As a result, we provide better bounds for the sampling complexity/number of low-rate analog-to-digital converters (ADCs) required to deterministically recover frequency-sparse wideband signals via DSFT in signal processing applications (Laska, 2006), (Kirolos et al., 2006).

Directional discrepancy in two dimensions

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.

A GENERALIZATION OF ROTH'S THEOREM IN FUNCTION FIELDS

Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let denote the subset of 𝔽q[t] containing all polynomials of degree strictly less than N. For non-zero elements r1, …, rs of 𝔽q satisfying r1 + ⋯ + rs = 0, let denote the maximal cardinality of a set which contains no non-trivial solution of r1x1 + ⋯ + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that .

Minimal Mahler Measure in Real Quadratic Fields

ABSTRACT We investigate the upper and lower bounds on the minimal Mahler measure of an irrational number lying in a particular real quadratic field.

A Prime Analogue of Roth’s Theorem in Function Fields

Let \(\mathbb{F}_{q}[t]\) denote the polynomial ring over the finite field \(\mathbb{F}_{q}\), and let \(\mathcal{P}_{R}\) denote the subset of \(\mathbb{F}_{q}[t]\) containing all monic irreducible polynomials of degree R. For non-zero elements r = (r1, r2, r3) of \(\mathbb{F}_{q}\) satisfying r1 + r2 + r3 = 0, let \(D(\mathcal{P}_{R}) = D_{\mathbf{r}}(\mathcal{P}_{R})\) denote the maximal cardinality of a set \(A_{R} \subseteq \mathcal{P}_{R}\) which contains no non-trivial solution of \(r_{1}x_{1} + r_{2}x_{2} + r_{3}x_{3} = 0\) with x i ∈ A R (1 ≤ i ≤ 3). By applying the polynomial Hardy-Littlewood circle method, we prove that \(D(\mathcal{P}_{R}) \ll _{q}\vert \mathcal{P}_{R}\vert /(\log \log \log \log \vert \mathcal{P}_{R}\vert )\).

RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS

Let k = Fq(t) be the rational function field over Fq and f(x) ∈ k[x1, . . . , xs] be a form of degree d. For l ∈ N, we establish that whenever s > l+ d ∑ w=1 w ( d− w + l − 1 l − 1 ) , the projective hypersurface f(x) = 0 contains a k-rational linear space of projective dimension l. We also show that if s > 1 + d(d+ 1)(2d+ 1)/6 then for any k-rational zero a of f(x) there are infinitely many s-tuples (

A note on compressed sensing and the complexity of matrix multiplication

1, . . . ,

The Manin conjecture for x0y0+⋯+xsys=0

s) of monic irreducible polynomials over k, with the

A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression

i not all equal, and f(a1