An important observation in compressed sensing is the exact recovery of an l0 minimiser to an underdetermined linear system via the l1 minimiser, given the knowledge that a sparse solution vector exists. Here, we develop a continuous analogue of this observation and show that the best L1 and L0 polynomial approximants of a corrupted function (continuous analogue of sparse vectors) are equivalent. We use this to construct best L1 polynomial approximants of corrupted functions via linear programming. We also present a numerical algorithm for computing best L1 polynomial approximants to general continuous functions, and observe that compared with best L-infinity and L2 polynomial approximants, the best L1 approximants tend to have error functions that are more localized.
Joint work with Alex Townsend (MIT).
Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about
$$\sum_1^N (a_{j+1}-a_j)^2?$$
When considering $E_k$ numbers (products of exactly $k$ primes), it is natural to ask, how they are distributed in short intervals. One can show much stronger results when one restricts to almost all intervals. In this context, we seek the smallest value of c such that the intervals $[x,x+(\log x)^c]$ contain an $E_k$ number almost always. Harman showed that $c=7+\varepsilon$ is admissible for $E_2$ numbers, and this was the best known result also for $E_k$ numbers with $k>2$.
We show that for $E_3$ numbers one can take $c=1+\varepsilon$, which is optimal up to $\varepsilon$. We also obtain the value $c=3.51$ for $E_2$ numbers. The proof uses pointwise, large values and mean value results for Dirichlet polynomials as well as sieve methods.
Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$.
The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.
This is joint work with Christopher Frei and Daniel Loughran.
The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.
For $k \geq 3$ we give new values of $\rho_k$ such that
$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$
has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean
value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also
depend on bounding the number of solutions of a congruence of the form
$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$
where $q$ is a given large natural number.
In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.
