On a paper of K. Soundararajan on smooth numbers in arithmetic progressions

In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\sqrt{e}-\delta} \geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1.Comment: 18 page

Monochromatic sums and products

Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.Comment: 48 pages, accepted for publication in Discrete Analysis. Second version has minor changes arising from the referee report. Third version updated to DAJ format. in Discrete Analysis 2016:

Inhomogeneous quadratic congruences

We investigate the density of integer solutions to certain binary inhomogeneous quadratic congruences and use this information to detect almost primes on a singular del Pezzo surface of degree 6.Comment: 24 page

Ding-Iohara-Miki symmetry of network matrix models

Ward identities in the most general "network matrix model" can be described in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation that such algebras and their various limits/reductions are the relevant substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q_1, q_2, q_3) deformed network matrix models. Exhaustive for these purposes should be the Pagoda triple-affine elliptic DIM, which corresponds to networks associated with 6d gauge theories with adjoint matter (double elliptic systems). We provide some details on elliptic qq-characters.Comment: 20 pages, 2 figure