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We study Piatetski-Shapiro sequences
$(\lfloor n^{c}\rfloor )_{n}$
modulo
$m$
, for non-integer
$c>1$
and positive
$m$
, and we are particularly interested in subword occurrences in those sequences. We prove that each block
$\in \{0,1\}^{k}$
of length
$k<c+1$
occurs as a subword with the frequency
$2^{-k}$
, while there are always blocks that do not occur. In particular, those sequences are not normal. For
$1<c<2$
, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence
$\lfloor n^{c}\rfloor$
modulo
$m$
is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.
In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke
$L$
-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
We study the analogue of the Bombieri–Vinogradov theorem for
$\operatorname{SL}_{m}(\mathbb{Z})$
Hecke–Maass form
$F(z)$
. In particular, for
$\operatorname{SL}_{2}(\mathbb{Z})$
holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on
$\operatorname{SL}_{2}(\mathbb{Z})$
, and
$\operatorname{SL}_{3}(\mathbb{Z})$
Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to
$1/2,$
which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for
$a\neq 0,$
where
$\unicode[STIX]{x1D70C}(n)$
are Fourier coefficients
$\unicode[STIX]{x1D706}_{f}(n)$
of a holomorphic Hecke eigenform
$f$
for
$\operatorname{SL}_{2}(\mathbb{Z})$
or Fourier coefficients
$A_{F}(n,1)$
of its symmetric-square lift
$F$
. Further, as a consequence, we get an asymptotic formula
where
$E_{1}(a)$
is a constant depending on
$a$
. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function
$\unicode[STIX]{x1D70C}(n)d(n-a)$
.
where
$[x]$
denotes the integral part of real
$x$
. The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].
We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.
We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.
By combining classical techniques together with two novel asymptotic identities derived in recent work by Lenells and one of the authors, we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of Re{u} and Re{v} and for a variety of sets of summation, as well as particular cases of Mordell-Tornheim double sums.
In this note we give a characterization of
$\ell ^{p}\times \cdots \times \ell ^{p}\rightarrow \ell ^{q}$
boundedness of maximal operators associated with multilinear convolution averages over spheres in
$\mathbb{Z}^{n}$
.
We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences
$\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$
with
$c>1$
a non-integral real number and
$k\in \mathbb{N}$
, as well as for
$\unicode[STIX]{x1D708}(p)$
where
$p$
runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.
The cardinality of the set of
$D\leqslant x$
for which the fundamental solution of the Pell equation
$t^{2}-Du^{2}=1$
is less than
$D^{1/2+\unicode[STIX]{x1D6FC}}$
with
$\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$
is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the
$q$
-analogue of van der Corput method to algebraic exponential sums with smooth moduli.
In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of
$\ell$
-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist
$x\in [1,p]$
and
$a\in \mathbb{F}_{p}^{\times }$
such that
$|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$
. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.
We study logarithmically averaged binary correlations of bounded multiplicative functions
$g_{1}$
and
$g_{2}$
. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever
$g_{1}$
or
$g_{2}$
does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions
$g_{j}$
, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of
$g_{1}$
and
$g_{2}$
is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of
$n$
and
$n+1$
are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdős and Pomerance on two consecutive smooth numbers. Thirdly, we show that if
$Q$
is cube-free and belongs to the Burgess regime
$Q\leqslant x^{4-\unicode[STIX]{x1D700}}$
, the logarithmic average around
$x$
of the real character
$\unicode[STIX]{x1D712}\hspace{0.6em}({\rm mod}\hspace{0.2em}Q)$
over the values of a reducible quadratic polynomial is small.
Let
$\Vert \cdots \Vert$
denote distance from the integers. Let
$\unicode[STIX]{x1D6FC}$
,
$\unicode[STIX]{x1D6FD}$
,
$\unicode[STIX]{x1D6FE}$
be real numbers with
$\unicode[STIX]{x1D6FC}$
irrational. We show that the inequality
has infinitely many solutions in primes
$p$
, sharpening a result due to Harman [On the distribution of
$\unicode[STIX]{x1D6FC}p$
modulo one II. Proc. Lond. Math. Soc. (3)72 (1996), 241–260] in the case
$\unicode[STIX]{x1D6FD}=0$
and Baker [Fractional parts of polynomials over the primes. Mathematika63 (2017), 715–733] in the general case.
We prove that the exponent of distribution of
$\unicode[STIX]{x1D70F}_{3}$
in arithmetic progressions can be as large as
$\frac{1}{2}+\frac{1}{34}$
, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double
$q$
-analogue of the van der Corput method for smooth bilinear forms.
We show that integral monodromy groups of Kloosterman
$\ell$
-adic sheaves of rank
$n\geqslant 2$
on
$\mathbb{G}_{m}/\mathbb{F}_{q}$
are as large as possible when the characteristic
$\ell$
is large enough, depending only on the rank. This variant of Katz’s results over
$\mathbb{C}$
was known by works of Gabber, Larsen, Nori and Hall under restrictions such as
$\ell$
large enough depending on
$\operatorname{char}(\mathbb{F}_{q})$
with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.
We establish bounds for triple exponential sums with mixed exponential and linear terms. The method we use is by Shparlinski [‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160] together with a bound for the additive energy from Roche-Newton et al. [‘New sum-product type estimates over finite fields’, Adv. Math.293 (2016), 589–605].
We prove that for any positive integers
$k,n$
with
$n>\frac{3}{2}(k^{2}+k+2)$
, prime
$p$
, and integers
$c,a_{i}$
, with
$p\nmid a_{i}$
,
$1\leqslant i\leqslant n$
, there exists a solution
$\text{}\underline{x}$
to the congruence
with
$1\leqslant {x_{i}\ll }_{k}p^{1/k}$
,
$1\leqslant i\leqslant n$
. This upper bound is best possible. Refinements are given for smaller
$n$
, and for variables restricted to intervals in more general position. In particular, for any
$\unicode[STIX]{x1D700}>0$
we give an explicit constant
$c_{\unicode[STIX]{x1D700}}$
such that if
$n>c_{\unicode[STIX]{x1D700}}k$
, then there is a solution with
$1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$
.
for infinitely many primes
$p$
that supersede those of Harman [Trigonometric sums over primes I. Mathematika28 (1981), 249–254; Trigonometric sums over primes II. Glasg. Math. J.24 (1983), 23–37] and Wong [On the distribution of
$\unicode[STIX]{x1D6FC}p^{k}$
modulo 1. Glasg. Math. J.39 (1997), 121–130].