相关论文: Roth's theorem in the primes
This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…
Combining Freiman's theorem with Balog-Szemeredi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In…
We establish unconditional $\Omega$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an…
The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…
We study the existence of products of primes in arithmetic progressions, building on the work of Ramar\'e and Walker. One of our main results is that if $q$ is a large modulus, then any invertible residue class mod $q$ contains a product of…
In this paper we prove: If 0 < d < 1, and p is a sufficiently large prime, then if S is a subset of Z/pZ having the least number of three-term arithmetic progressions among all subsets of Z/pZ having at least dp elements, then S has an…
The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic…
We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.
In this article, we prove a theorem \`a la Mauduit et Rivat (prime number theorem, Moebius randomness principle) for functions that count digital blocks whose length is a growing function tending to infinity. These sequences are not…
We show that once $\theta>17/30$, every sufficiently long interval $[x,x+x^\theta]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for each fixed $k\ge3$ and $\theta>17/30$,…
A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to…
A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…
We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the…
We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…
In this paper, we generalize Mauduit and Rivat's theorem on the Rudin-Shapiro sequence. Weakening the hypothesis needed in their theorem, we prove a prime number theorem for a large class of functions defined on the digits. Our result…
The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general…
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…
We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a…