On The Hereditary Discrepancy of Homogeneous Arithmetic Progressions

We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by $n^{1/O(\log \log n)}$. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound on the discrepancy of set systems of subcubes of the boolean cube $\{0, 1\}^d$.Comment: To appear in the Proceedings of the American Mathematical Societ

Arithmetic properties of Fredholm series for p-adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.Comment: Final version. Numbering in main body different different from previous version. To appear in Proc. Lon. Math. Soc. 25 pages, 7 table

Arithmetic properties of Fredholm series for -adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent -adic modular forms "near the boundary" of -adic weight space. We also prove in tame level 1 that the coeffcients of the Fredholm series of the U operator never vanish modulo , a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo using a modular interpretation of the mod reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.Accepted manuscrip

Polignac Numbers, Conjectures of Erd\"os on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and Zhang, respectively

Shifted distinct-part partition identities in arithmetic progressions

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $p_{S_1}(n-H) = p_{S_2}(n)$ for all $n$ in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi's theorem to other arithmetic progressions

Slopes of modular forms and the ghost conjecture

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".Comment: 17 pages. 2 figures. Minor changes from v1. Final version. To appear in IMRN. arXiv admin note: text overlap with arXiv:1607.0465