Mathematics

Let A be a set of finite integers, define A+A \ = \ \{a_1+a_2: a_1,a_2 \in A\}, \ \ \ A-A \ = \ \{a_1-a_2: a_1,a_2 \in A\}, and for non-negative integers s and d define sA-dA\ =\ \underbrace{A+\cdots+A}_{s} -\underbrace{A-\cdots-A}_{d}. A More Sums than Differences (MSTD) set is an A where |A+A| > |A-A| . It was initially thought that the percentage of subsets of [0,n] that are MSTD would go to zero as n approaches infinity as addition is commutative and subtraction is not. However, in a surprising 2006 result, Martin and O'Bryant proved that a positive percentage of sets are MSTD, although this percentage is extremely small, about 10^{-4} percent. This result was extended by Iyer, Lazarev, Miller, ans Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for \{s_1,d_1\} \neq \{s_2, d_2\} and s_1+d_1=s_2+d_2 with |s_1A-d_1A| > |s_2A-d_2A| , and that in d -dimensions, a positive percentage of sets are MSTD. For many such results, establishing explicit MSTD sets in 1 -dimensions relies on the specific choice of the elements on the left and right fringes of the set to force certain differences to be missed while desired sums are attained. In higher dimensions, the geometry forces a more careful assessment of what elements have the same behavior as 1 -dimensional fringe elements. We study fringes in d -dimensions and use these to create new explicit constructions. We prove the existence of generalized MSTD sets in d -dimensions and the existence of k -generational sets, which are sets where |cA+cA|>|cA-cA| for all 1\leq c \leq k . We then prove that under certain conditions, there are no sets with |kA+kA|>|kA-kA| for all k \in \mathbb{N}.

We call R G (a):= ∑ ∞ q=1 G(q) c q (a) the 'Ramanujan series', of coefficient G: N → C, where c q (a) is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define G a 'Ramanujan coefficient' when R G (a) converges pointwise, in all natural a . Then, R G : N → C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed F: N → C is <F>:= { G:N→C| R G w−d,F= R G }. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients G : their <F> subset is called the 'multiplicative Ramanujan cloud', <F > M . Our first main result, the "Finiteness convergence Theorem", for G multiplicative, among other properties equivalent to " R G well defined", reduces the convergence test to a finite set, i.e., R G w-d is equivalent to: R G (a) converges for all a dividing N(G)∈ N, that we call the "Ramanujan conductor". Our second main result, the "Finite Euler product explicit formula", for multiplicative Ramanujan coefficients G , writes F= R G as a finite Euler product; thus, F is a semi-multiplicative function (following Rearick definition) and this product is the Selberg factorization for F . In particular, we have: F(a)= R G (a) converges absolutely, being finite (of length depending on non-zero p− adic valuations of a ). Our third main result, called the "Multiplicative Ramanujan clouds", studies the important subsets of <F > M ; also giving, for all multiplicative F , the 'canonical Ramanujan coefficient' G F ∈<F > M , proving: Any multiplicative F has a finite Ramanujan expansion with multiplicative coefficients.

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic x 2 + y 2 + z 2 ??xyz=0. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. One can associate to each a positive rational number a Markov number in a natural way. We give a new unified proof of certain conjectures from Martin Aigner's book, Markov's Theorem and 100 Years of the Uniqueness Conjecture. Our proof relies on a relationship between Markov numbers and the lengths of closed simple geodesics on the punctured torus discovered by H. Cohn.

We prove analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers n= p 1 p 2 ?�X such that n+h is a product of exactly two primes which holds for almost all |h|?�H with log 19+ε X?�H??X 1?��?, under a restriction on the size of p 1 . Additionally, we consider correlations n,n+h where n is a prime and n+h has exactly two prime factors, establishing an asymptotic formula which holds for almost all |h|?�H with X 1/6+ε ?�H??X 1?��?.

We develop an approach to study character sums, weighted by a multiplicative function f: F q [t]→ S 1 , of the form ∑ G∈ M N f(G)χ(G)ξ(G), where χ is a Dirichlet character and ξ is a short interval character over F q [t]. We then deduce versions of the Matomäki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields F q [t] , where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of q . Compared with the integer setting, we encounter some different phenomena, specifically a low characteristic issue in the case that q is a power of 2 , as well as the need for a wider class of ''pretentious" functions called Hayes characters. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case. In a companion paper, we will use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve the ''corrected" form of the Erdős discrepancy problem over F q [t] .

In this paper we use computational methods to disprove a conjecture by Alaoglu and Erdős regarding the superabundant numbers.

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.

We find upper and lower bounds on the number of rational points that are ? -approximations of some n -dimensional p -adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of p -adic weighted simultaneously approximable points intersected with p -adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that ? -approximate a p -adic integer form a set of resonant points that can be used to construct a local ubiquitous system of rectangles.

We explore the covariance of error terms coming from Weyl's conjecture regarding the number of Dirichlet eigenvalues up to size X . We also consider this problem in short intervals, i.e. the error term of the number of eigenvalues in the window [X,X+S] for some S(X) . We look at these error terms for planar domains where the Dirichlet eigenvalues can be explicitly calculated. In these cases, the error term is closely related to the error term from the classical lattice points counting problem of expanding planar domains. We give a formula for the covariance of such error terms, for general planar domains. We also give a formula for the covariance of error terms in short intervals, for sufficiently large intervals. Going back to the Dirichlet eigenvalue problem, we give results regarding the covariance of the error terms in short intervals of 'generic' rectangles. We also explore a specific example, namely we compute the covariance between the error terms of an equilateral triangle and various rectangles.

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves X 0 (N) for N?�{53,57,61,65,67,73} as well as the quartic points on X 0 (65) . To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on X 0 (65) , we rigorously compute the full rational Mordell--Weil group of its Jacobian.