Related papers: Bihomogeneous forms in many variables
Let U be a homogeneous variety over Q of a linear algebraic group. Choose an integral model and assume the existence of infinitely many integral points. Then one would like to give an asymptotic count of integral points of bounded height…
We use the circle method to count the number of integer solutions to systems of bihomogeneous equations of bidegree $(1,1)$ and $(2,1)$ of bounded height in lopsided boxes. Previously, adjusting Birch's techniques to the bihomogeneous…
We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method…
We give a variant of Weyl's inequality for systems of forms together with applications. First we use this to give a different formulation of a theorem of B. J. Birch on forms in many variables. More precisely, we show that the dimension of…
We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences…
In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems as in [AR]. We give some suffi cient conditions ensuring their validity, related to the…
Littlewood's theorem is one of the pioneering results in random analytic functions over the open unit disk. In this paper, we prove some analogues of this theorem for Hardy spaces in infinitely many variables. Our results not only cover…
The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the…
We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the…
We give a revised version of Schmidt's treatment of forms in many variables, which allows us to prove a Hasse principle under more lenient conditions on the number of variables than what had previously been thought possible with these…
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to…
We show how to adapt the Hardy--Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations…
Suppose that we are interested in the comparison of two independent categorical variables. Suppose also that the population is divided into subpopulations or groups. Notice that the distribution of the target variable may vary across…
This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates…
We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local…
We present new estimate for Hardy-type inequality in variable exponent Lebesgue spaces. More precisely, by imposing regularity assumptions on the exponent, we prove that the estimations can be reduced to the fixed exponents.
This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood.…
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…
We give a counting formula in terms of modified Hall-Littlewood polynomials and the chromatic quasisymmetric function for the number of points on an arbitrary Hessenberg variety over a finite field. As a consequence, we express the…
When we use the normal mixture model, the optimal number of the components describing the data should be determined. Testing homogeneity is good for this purpose; however, to construct its theory is challenging, since the test statistic…