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Suppose \(N\) is sufficiently large and \(A\subset [N^{1-1/\log \log N},N]\) is such that

\(R(A)\geq 2(\log N)^{1/500}\),

every \(n\in A\) is divisible by some prime \(p\) satisfying \(5 \leq p \leq (\log N)^{1/500}\),

every prime power \(q\) dividing some \(n\in A\) satisfies \(q\leq N^{1-6/\log \log N}\), and

every \(n\in A\) satisfies

\[ \tfrac {99}{100}\log \log N\leq \omega (n) \leq 2\log \log N. \]

There is some \(S\subset A\) such that \(R(S)=1\).

For any finite \(A\subset \mathbb {N}\) and prime power \(q\) we define

and let \(\mathcal{Q}_A\) be the set of all prime powers \(q\) such that \(A_q\) is non-empty (i.e. those \(p^r\) such that \(p^r\| n\) for some \(n\in A\)).

For any finite set of naturals \(A\) and integer \(k\geq 1\), and real \(K{\gt}0\), we define the ‘major arc’ corresponding to \(t\in \mathbb {Z}\) as

Let \(\mathfrak {M}(A,k,K) = \cup _{t\in \mathbb {Z}}\mathfrak {M}(t;A,k,K)\).

Let \(I_h(K,k)\) be the interval of length \(K\) centred at \(kh\), and let \(\mathfrak {m}_1(A,k,K,\delta )\) be those \(h\in J(A)\backslash \mathfrak {M}(A,k,K)\) such that

Let \(\mathfrak {m}_2(A,k,K,\delta )\) be the rest of \(J(A)\backslash \mathfrak {M}(A,k,K)\).

Suppose \(N\) is sufficiently large and \(N\geq M\geq N^{1/2}\), and suppose that \(A\subset [M,N]\) is a set of integers such that

Then for any interval of length \(\leq MN^{-2/(\log \log N)}\), if \(A_I\) is the set of those \(n\in A\) which divide some element of \(I\), for all \(q\in \mathcal{Q}_A\) such that \(R(A_I;q){\gt} 1/2(\log N)^{1/100}\) there exists some integer \(x_q\in I\) such that \(q\mid x_q\) and

Suppose \(N\) is an integer and \(\delta ,M\in \mathbb {R}\). Suppose that \(A\subset [M,N]\) is a set of integers such that, for all \(q\in \mathcal{Q}_A\),

Then for any finite set of integers \(I\), for all \(q\in \mathcal{D}_I(A;\delta M)\), if \(A_I\) is the set of those \(n\in A\) that divide some element of \(I\),

Let \(A\) be a finite set of natural numbers not containing \(0\). If

and for all \(p\in \mathcal{P}_A\) then \(r_p\geq 0\) is the greatest integer such that \(p^{r_p}\) divides some \(n\in A\), then

Let \(M\geq 1\) and \(A\) a finite set of naturals such that \(n\geq M\) for all \(n\in A\). Let \(K\) be a real such that \(K{\lt}M\). Let \(k\geq 1\) be an integer which divides \([A]\). Suppose that \(kR(A) \in [2-k/M,2)\).

Suppose that \(N\) is sufficiently large and \(M\geq 8\). Let \(T{\gt}0\) be a real and \(k\geq 1\) be an integer. Let \(A\subset [M,N]\) be a set of integers such that if \(q\in \mathcal{Q}_A\) then \(q\leq \frac{TK^2}{N^2 \log N}\). Then

Let \(N\) be a sufficiently large integer. Let \(K,L{\gt}0\) be reals such that \(0{\lt}K\leq N\). Let \(T{\gt}0\) be any real. Let \(k\) be an integer such that \(1\leq k \leq N/64\). Let \(A\subset [1,N]\) be a finite set of integers such that

if \(q\in \mathcal{Q}_A\) then \(q\leq \tfrac {1}{16}\frac{LK^2}{N^2(\log N)^2}\), and

for any interval \(I\) of length \(K\), either

- \[ \# \{ n\in A : \textrm{no element of }I\textrm{ is divisible by }n\} \geq T, \]
or

there is some \(x\in I\) divisible by all \(q\in \mathcal{D}_I(A;L)\).

Then

Suppose that \(N\geq 4\). Let \(A\subset [1,N]\) be a finite set of integers and \(t\) an integer. Let \(K,L{\gt}0\) be reals and suppose that \(q\leq \tfrac {1}{16}LK^2/N^2(\log N)^2\) for all \(q\in \mathcal{Q}_A\). Let \(\mathcal{D}=\mathcal{D}_I(A;L)\) where \(I\) is the interval of length \(K\) centred at \(t\). Then

Let \(M\geq 1\) and \(A\) a finite set of naturals such that \(n\geq M\) for all \(n\in A\). Let \(K\) be a real such that \(K{\lt}M\). Let \(k\geq 1\) be an integer which divides \([A]\). Suppose that \(kR(A) \in [2-k/M,2)\), and there is no \(S\subset A\) such that \(R(S)=1/k\), and \([A]\leq 2^{\left\lvert A\right\rvert -1}\). Then

For any \(X\geq 3\)

If \(k\geq 1\) is an integer and \(A\) is a finite set of natural numbers such that there is no \(S\subset A\) such that \(R(S)=1/k\), and \(R(A){\lt}2/k\), and \([A]\leq 2^{\left\lvert A\right\rvert -1}\) then

Suppose that \(N\) is sufficiently large and \(N{\gt}M\). Let \(\epsilon ,\alpha \) be reals such that \(\alpha {\gt} 4\epsilon \log \log N\) and \(A\subset [M,N]\) be a set of integers such that

and if \(q\in \mathcal{Q}_A\) then \(q\leq \epsilon M\).

There is a subset \(B\subset A\) such that \(R(B)\in [\alpha -1/M,\alpha )\) and, for all \(q\in \mathcal{Q}_B\), we have \(\epsilon {\lt} R(B;q)\).

Let \(1/2{\gt}\epsilon {\gt}0\) and \(N\) be sufficiently large, depending on \(\epsilon \). If \(A\) is a finite set of integers such that \(R(A)\geq (\log N)^{-\epsilon /2}\) and \((1-\epsilon )\log \log N\leq \omega (n)\leq 2\log \log N\) for all \(n\in A\) then

Let \(N\) be sufficiently large and \(z,y\) be two parameters such that \(\log N \geq z{\gt}y\geq 3\). If \(X\) is the set of all those integers not divisible by any prime in \(p\in [y,z]\) then

Let \(N\) be sufficiently large and \(z,y\) be two parameters such that \((\log N)^{1/2}\geq z{\gt}4y\geq 8\). If \(Y\subset [1,N]\) is the set of all those integers divisible by at least two distinct primes \(p_1,p_2\in [y,z]\) where \(4p_1{\lt}p_2\) then

For any \(x,y\geq 0\) and \(u\geq v\geq 1\)

There is a constant \(C{\gt}0\) such that the following holds. Let \(0{\lt}y{\lt}N\) be some reals, and suppose that \(D\) is a finite set of integers such that if \(d\in D\) and \(p^r\| d\) then \(y{\lt}p^r\leq N\). Then, for any real \(k\geq 1\),

Let \(N\) be sufficiently large. Let \(\epsilon ,y,w,M\) be reals such that \(M{\lt}N\) and \(2\leq \lceil y\rceil \leq \lfloor w\rfloor \) and \(1/M{\lt} \epsilon \log \log N\) and \(\frac{2}{w^2}\geq 3\epsilon \log \log N\) and let \(A\subset [M,N]\) be such that

\(R(A)\geq 2/y+2\epsilon \log \log N\),

every \(n\in A\) is divisible by some \(d\in [y,w]\),

if \(q\in \mathcal{Q}_A\) then \(q\leq \epsilon M\).

Then there exists some \(\emptyset \neq A'\subseteq A\) and \(d\in [y,w]\) such that \(R(A')\in [2/d-1/M,2/d)\), for all \(q\in \mathcal{Q}_{A'}\) we have \(R(A';q){\gt}\epsilon \), and there is a multiple of \(d\) in \(A'\), and there are no multiples of any \(m\in [y,d)\) in \(A\).

There is a constant \(c{\gt}0\) such that the following holds. Let \(N\geq M\geq N^{1/2}\) be sufficiently large, and suppose that \(k\) is a real number such that \(1\leq k \leq c\log \log N\). Suppose that \(A\subset [M,N]\) is a set of integers such that \(\omega (n)\leq (\log N)^{1/k}\) for all \(n\in A\).

For all \(q\) such that \(R(A;q)\geq 1/\log N\) there exists \(d\) such that

\(qd {\gt} M\exp (-(\log N)^{1-1/k})\),

\(\omega (d)\leq \tfrac {5}{\log k}\log \log N\), and

- \[ \sum _{\substack {n\in A_q\\ qd\mid n\\ (qd,n/qd)=1}}\frac{qd}{n}\gg \frac{R(A;q)}{(\log N)^{2/k}}. \]

There exists a constant \(c{\gt}0\) such that the following holds. Suppose that \(N\) is sufficiently large. Let \(K,L,M,T\) be reals such that \(T,L{\gt}0\) and \(8\leq K{\lt}M\leq N\), and let \(k\) be an integer such that \(1\leq k\leq M/64\). Let \(A\subset [M,N]\) be a set of integers such that

\(R(A)\in [2/k-1/M,2/k)\),

\(k\) divides \([A]\),

if \(q\in \mathcal{Q}_A\) then \(q\leq c\min \left( M/k,\frac{LK^2}{N^2(\log N)^2},\frac{TK^2}{N^2\log N}\right)\), and

for any interval \(I\) of length \(K\), either

- \[ \# \{ n\in A : \textrm{no element of }I\textrm{ is divisible by }n\} \geq T, \]
or

there is some \(x\in I\) divisible by all \(q\in \mathcal{D}_I(A; L)\).

There is some \(S\subset A\) such that \(R(S)=1/k\).

Suppose \(N\) is sufficiently large and \(N\geq M\geq N^{1/2}\), and suppose that \(A\subset [M,N]\) is a set of integers such that

and, for all \(q\in \mathcal{Q}_A\),

Then either

there is some \(B\subset A\) such that \(R(B)\geq \tfrac {1}{3}R(A)\) and

\[ \sum _{q\in \mathcal{Q}_{B}}\frac{1}{q}\leq \frac{2}{3}\log \log N, \]or

for any interval of length \(\leq MN^{-2/(\log \log N)}\), either

- \[ \# \{ n\in A : \textrm{no element of }I\textrm{ is divisible by }n\} \geq M/\log N, \]
or

there is some \(x\in I\) divisible by all \(q\in \mathcal{D}_I(A;M/2(\log N)^{1/100})\).

Suppose \(N\) is sufficiently large and \(N\geq M\geq N^{1/2}\), and suppose that \(A\subset [M,N]\) is a set of integers such that

and, for all \(q\in \mathcal{Q}_A\),

and \(\sum _{q\in \mathcal{Q}_A}\frac{1}{q}\leq \frac{2}{3}\log \log N\). Then for any interval of length \(\leq MN^{-2/(\log \log N)}\), either

- \[ \# \{ n\in A : \textrm{no element of }I\textrm{ is divisible by }n\} \geq M/\log N, \]
or

there is some \(x\in I\) divisible by all \(q\in \mathcal{D}_I(A;M/2(\log N)^{1/100})\).

Let \(N\) be sufficiently large. Suppose \(A\subset [N^{1-1/\log \log N},N]\) and \(1\leq y\leq z\leq (\log N)^{1/500}\) are such that

\(R(A)\geq 2/y+(\log N)^{-1/200}\),

every \(n\in A\) is divisible by some \(d_1\) and \(d_2\) where \(y\leq d_1\) and \(4d_1\leq d_2\leq z\),

every prime power \(q\) dividing some \(n\in A\) satisfies \(q\leq N^{1-6/\log \log N}\), and

every \(n\in A\) satisfies

\[ \tfrac {99}{100}\log \log N\leq \omega (n) \leq 2\log \log N. \]

There is some \(S\subset A\) such that \(R(S)=1/d\) for some \(d\in [y,z]\).

There is a constant \(C{\gt}0\) such that the following holds. If \(A\subset \{ 1,\ldots ,N\} \) and

then there is an \(S\subset A\) such that \(\sum _{n\in S}\frac{1}{n}=1\).