On the Decay of the Fourier Transform and Three Term Arithmetic Progressions
Ernie Croot
备注：
One small notational correction: In the paper I called ||f||_(1/3) a `norm', when in fact it should be 'quasinorm'. This does not affect any results, as I don't use the triangle inequality anywhere -- the 1/3 quasinorm was only used as a convenient way to state a corollary of one of my results
In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-δin place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too ``small'', then there are lots of triples m,m+d,m+2d such that f(m)f(m+d)f(m+2d) > 0. If f is the indicator function for some set S, then this would be saying that the set has many three-term arithmetic progressions. In principle this theorem can be applied to sets having very low density, where |S| is around p^{n(1-c)} for some small c > 0.
We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.
We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of arithmetic progressions that satisfy additional congruence conditions and provide asymptotic approximations.
We prove that if $A\subseteq \{1,\dots,N\}$ does not contain any non-trivial three-term arithmetic progression, then $$|A|\ll \frac{(\log\log N)^{3+o(1)}}{\log N}N\,.$$
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
We answer a number of questions of Erdős on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterlé.
We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.
We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.
An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.
We prove an explicit error term for the $ψ(x,χ)$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.
We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.
We show estimates for the distribution of $k$-free numbers in short intervals and arithmetic progressions. We argue that, at least in certain ranges, these estimates agree with a conjecture by H. L. Montgomery.
In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_εN^{1/2 - ε}$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d \ll N^{1/2}$ unconditionally. We also prove some partial results when $k \ge 4$ and pose some open questions.
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.
