Abstract We describe some corrections to be made to our previous paper. The main result is not affected.

Let 1 < c < 24/19. We show that the number of integers n <= N that cannot be written as [p(1)(c)] + [p(2)(c)] (p(1), p(2) primes) is O(N1-sigma+epsilon). Here s is a positive function of c (given explicitly) and epsilon is an arbitrary positive number.

We obtain new estimates on the maximal operator applied to the Weyl sums. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).

Let 1< c < 24/19 . We show that the number of integers n ≤ N that cannot be written as [p_1^c] + [p_2^c] ( p_1 , p_2 primes) is O(N^1-σ +ε) . Here σ is a positive function of c (given explicitly) and ε is an arbitrary positive number.

In this paper we give a refinement of the one-dimensional form of Bilu’s Equidistribution Theorem together with some examples and applications. These include an explicit bound for the sum of the $e^{i\theta }$ taken over all conjugates $re^{i\theta }$ of an algebraic number $\alpha $ in terms of its height $h$ and degree $d$. That is applied to show that a certain discrepancy of $\alpha $ is at most $8000h^{1/3}$, which removes an extra term involving $d$ from previous bounds; this cannot be done simply by using a lower bound for $h$ in terms of $d$, even assuming the Lehmer Conjecture. And we give a new estimate for the norm of $1-\alpha $. We also improve existing upper bounds for the height of $\xi $ when $\xi ,1-\xi $ are multiplicatively dependent.

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials f(X)=Xk$f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K. Halupczok (2012, 2015, 2018) and M. Munsch (2020). We also consider moduli defined by polynomials f(X)∈Z[X]$f(X) \in \mathbb {Z}[X]$ , Piatetski–Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri–Vinogradov theorem with Piatetski–Shapiro moduli improving the level of distribution of R. C. Baker (2014).

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given α∈(1/2,1) achieve the order at least Nα for infinitely many sum lengths N. For Weyl sums with polynomials of degree d⩾3 we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving the exact value for the corresponding Hausdorff dimension when α is close to 1. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.

Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.

The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many solutions in primes exist.

An asymptotic formula is given for the number of y-smooth numbers up to x in a Beatty sequence corresponding to an irrational number of finite type.

Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn + tn^2$. Suppose that $|S(x,t)|$ attains its maximum for given $x$ at $t = t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of $S(x,t(x))$.

We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth numbers tailored to the application. The new exponent approaches 1/3 as the exponent of log n becomes large.

If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.

The well‐known Siegel Lemma gives an upper bound cUm/(n−m) for the size of the smallest non‐zero integral solution of a linear system of m⩾1 equations in n>m unknowns whose coefficients are integers of absolute value at most U⩾1 ; here c=c(m,n)⩾1 . In this paper, we show that a better upper bound Um/(n−m)/B is relatively rare for large B⩾1 ; for example, there are θ=θ(m,n)>0 and c′=c′(m,n) such that this happens for at most c′Umn/Bθ out of the roughly (2U)mn possible such systems.

We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted to the sequences of integer parts $[F(n)]$, where $F(t) = t^c$ ($c > 1$, $c \not\in \mathbb{N}$) or $F(t) = \exp\big((\log t)^{\gamma}\big)$ ($1 < \gamma < 3/2$).

Let f be a polynomial with irrational leading coefficient. We obtain inequalities for the distance from the nearest integer of f(p) that hold for infinitely many primes p. These results improve work of Harman in 1981 and 1983 and Wong in 1997.

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+\epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\epsilon}$.

By combining a sieve method of Harman with the work of Maynard and Tao we show that lim inf _n→∞ (p_n+m-p_n)≪exp (3.815m).

Let d(n) := p(n+1) - p(n), where pn denotes the nth smallest prime, and let R(T) := log T log(2) T log(4) T/(log(3) T)(2) (the 'Erdos-Rankin' function). We consider the sequence (d(n)/R(p(n))) of normalized prime gaps, and show that its limit point set contains at least 25% of non-negative real numbers. We also show that the same result holds if R(T) is replaced by any 'reasonable' function that tends to infinity more slowly than R(T) log(3) T. We also consider 'chains' of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between primes with subsequent developments of Ford, Green, Konyagin, Maynard and Tao on long gaps between consecutive primes.