We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form $\{m,m+[p_n^a], m+[p_n^b]\}$ , where $a,b$ are positive nonintegers and $p_n$ denotes the nth prime, a property that fails if a or b is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences, and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime k-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.

We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.

Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum

$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$

$p$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}$

$k\in \mathbb{Z}$

$g:\mathbb{N}\rightarrow \mathbb{Z}$

$b$

$\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$

We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log x)^{{\it\epsilon}}$ containing $\gg _{{\it\epsilon}}\log \log x$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.

We discuss how much space is sufficient to decide whether a unary given numbern is a prime. We show thatO(log log n) space is sufficient for a deterministicTuring machine, if it is equipped with an additional pebble movable along the input tape,and also for an alternating machine, if the space restriction applies only to itsaccepting computation subtrees. In other words, the language is a prime is inpebble–DSPACE(log log n) and also inaccept–ASPACE(log log n). Moreover, if the givenn is composite, such machines are able to find a divisor ofn. Since O(log log n) space is toosmall to write down a divisor, which might requireΩ(log n) bits, the witness divisor is indicated by theinput head position at the moment when the machine halts.

It is shown that every sufficiently large integer congruent to $14$ modulo $240$ may be written as the sum of $14$ fourth powers of prime numbers, and that every sufficiently large odd integer may be written as the sum of $21$ fifth powers of prime numbers. The respective implicit bounds $14$ and $21$ improve on the previous bounds $15$ (following from work of Davenport) and $23$ (due to Thanigasalam). These conclusions are established through the medium of the Hardy-Littlewood method, the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve, rather than the conventional methods which `waste' a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers. In the work on fifth powers, a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone. 2000 Mathematics Subject Classification: 11P05, 11N36, 11L15, 11P55.