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Mathematics
G. A. Freiman
INVERSE PROBLEMS OF ADDITIVE NUMBER THEORY
On the Addition of Sets of Residues Modulo a Prime
(Presented by Academician I. M. Vinogradov, June 30, 1961)
Notation. \(K\) and \(M\) are finite sets of rational integers:
\[ K=\{a_0,a_1,\ldots,a_{k-1}\},\qquad M=\{b_0,b_1,\ldots,b_{m-1}\}. \]
We assume that \(a_0=0,\ a_i<a_{i+1},\ i=0,1,\ldots,k-2;\ b_0=0,\ b_j<b_{j+1},\ j=0,1,\ldots,m-2;\ a_{k-1}<p\) and \(b_{m-1}<p;\ p\) is a prime number. \((2K)_p,\ (K+M)_p\) are the sets of all distinct residues modulo \(p\) from \(2K,\ K+M\). \(T(2K), T(K+M)\) are the number of numbers in \(2K,\ K+M\). \(T_p(2K), T_p(K+M)\) are the number of distinct residues in \(2K,\ K+M\) modulo \(p\).
Results describing the structure of \(K\) and \(M\) when \(T_p(K+M)=k+m-1\) were obtained by Vosper \((^1)\).
Theorem 1. If \(T_p(2K)<2.4k-3\) and \(k<p/35\), then the residues from \(K\) are contained in an arithmetic progression modulo \(p\) of length \(k+b\), where \(b\) is determined from the equality \(T=2k-1+b\).
Proof. Let
\[ I=\sum_{x_1,x_2\in K}\sum_{x_3\in(2K)_p}\sum_{a=0}^{p-1} e^{2\pi i a\frac{x_1+x_2-x_3}{p}} = \sum_{a=0}^{p-1}S_1^2S, \]
where
\[ S_1=\sum_{j=0}^{k-1}e^{2\pi i\frac{a}{p}a_j},\qquad S=\sum_{x\in(2K)_p}e^{-2\pi i\frac{a}{p}x}. \]
If one assumes that for \(a\ne0\)
\[ |S_1|\leq 0.6k, \]
then
\[ |I|\leq k^2T_p+\sum_{a=1}^{p-1}|S_1|^2|S| \leq k^2T_p+0.6k \left[ \sum_{a=0}^{p-1}|S_1|^2 \sum_{a=0}^{p-1}|S|^2 \right]^{1/2} <k^2p. \]
Since \(I=k^2p\), there exists \(a'\) such that
\[ |S_1(a')|= \left| \sum_{j=0}^{k-1}e^{2\pi i\frac{a'}{p}a_j} \right| >0.6k. \tag{1} \]
Lemma. Given \(k\) complex numbers
\[ e^{2\pi i\alpha_0},\ e^{2\pi i\alpha_1},\ldots,\ e^{2\pi i\alpha_{k-1}}, \]
where \(\alpha_i\) are real numbers.
Let \(k_1(\beta)\) be the number of numbers for which
\[ \begin{array}{ll} \beta \leqslant \{\alpha_i\}<\beta+\tfrac12, & \text{if } 0\leqslant \beta\leqslant \tfrac12,\\ \beta \leqslant \{\alpha_i\}\ \text{or}\ \{\alpha_i\}<\beta-\tfrac12, & \text{if } \tfrac12<\beta<1. \end{array} \]
If for every \(\beta\)
\[ k_1(\beta)<ck, \]
then
\[ \left|\sum_{j=0}^{k-1} e^{2\pi i\alpha_j}\right|\leqslant (2c-1)k. \]
From this lemma and from (1) it follows that there exist integers \(u\) and \(v\) (the latter may be determined by means of the congruence \(a'v\equiv 1\pmod p\)) such that among the numbers
\[ u+vs,\qquad 0\leqslant s\leqslant \frac{p-3}{2} \tag{2} \]
there are \(k_1\) numbers congruent to numbers of \(K\), and
\[ k_1\geqslant 0.8k. \]
Let the numbers from (2) that are congruent to numbers of \(K\) be obtained for the values \(s\) equal to
\[ s_0,s_1,\ldots,s_{k_1-1},\qquad s_i<s_{i+1},\quad i=0,1,\ldots,k_1-2. \]
We may assume that \(s_0=0\).
If \((s_1,s_2,\ldots,s_{k_1-1})=d>1\), then instead of the number \(v\) one may take the number \(vd\). We may therefore suppose that \(d=1\).
If \(s_{k_1-1}\geqslant 2k_1-2\), then from Theorem VI in (²) it follows that
\[ T_p(2K)\geqslant 3k_1-3\geqslant 2.4k-3. \]
Thus, \(s_{k_1-1}\leqslant 2k_1-3\). If in \(K\) there existed a residue congruent to \(u+vs\) for
\[ 4k_1-6<s<p-(2k_1-3), \]
then we would have \(T_p\geqslant 2k_1-1+k_1=3k_1-1\).
Thus, all residues from \(K\) are congruent to the numbers \(u+vs\) for
\[ -(2k_1-3)\leqslant s\leqslant 4k_1-6. \]
Hence, and from Theorem V in (²), follows the validity of the theorem being proved.
The theorem is also valid for \(T_p<2.3k-3\) and \(k<p/12\).
Theorem 2. If \(3k-3\leqslant T(2K)<\tfrac{10}{3}k-5\), \((a_1,a_2,\ldots,a_{k-1})=1\), \(c_1k<a_{k-1}\), \(k>c_2\), where \(c_1\) and \(c_2\) are sufficiently large positive constants, then the set \(K\) is contained in two arithmetic progressions with the same difference, of total length not exceeding \(k+b\), where \(b\) is determined from the equality \(T=3k-3+b\) (see (³)).
Using this result instead of Theorem VI in (²), it is easy to strengthen the result of Theorem 1.
Theorem 3. If \(T_p<2.68k\), then there exist positive numbers \(\beta\) and \(c\) such that, when the condition \(c<k<\beta p\) is satisfied, the assertion of Theorem 1 is valid.
Theorem 4. If \(T=k+m-1+b\), where \(b\leqslant \min(k,m)-3\), \((a_1,a_2,\ldots,a_{k-1}, b_1,b_2,\ldots,b_{m-1})=1\), then \(a_{k-1}\leqslant k+b-1\), \(b_{m-1}\leqslant m+b-1\).
Theorem 5. If \(k\geqslant m\),
\[ T_p(K+M)\leqslant k+m+\theta m-3,\qquad k<\beta p,\quad \beta\leqslant \tfrac{1}{12}, \tag{3} \]
where \(\theta \geqslant 0\) satisfies the condition
\[ \eta^3\theta^3+(\eta^3+2\eta^2)\theta^2+ \left(\eta^2+\frac{5}{4}\eta\right)\theta+ \frac{\eta+1}{4}-2(1-3\beta)^2\eta^{3/2}<0, \qquad \eta=m/k \tag{4} \]
then the sets \(K\) and \(M\) are situated in progressions with the same difference modulo \(p\), of lengths respectively equal to \(k+b\) and \(m+b\), where \(b\) is determined from the equality \(T=k+m-1+b\).
Proof. Suppose
\[ |S_1S_2|<\gamma km, \]
where
\[ \gamma=(1-3\beta)^2\frac{\sqrt{km}}{k+m+\theta m},\qquad S_1=\sum_{j=0}^{k-1} e^{2\pi ia\frac{a_j}{p}}, \qquad S_2=\sum_{j=0}^{m-1} e^{2\pi ia\frac{b_j}{p}}, \qquad p\nmid a, \]
leads to a contradiction:
\[ \begin{aligned} pkm &= \sum_{x_1\in K}\sum_{x_2\in M}\sum_{x_3\in (K+M)_p} \sum_{a=0}^{p-1} e^{\frac{2\pi i a}{p}(x_1+x_2-x_3)} = \sum_{a=0}^{p-1} S_1S_2S_3 < \\ &< kmT+ \left[ \sum_{a=1}^{p-1}|S_1|^2|S_2|^2 \sum_{a=0}^{p-1}|S_3|^2 \right]^{1/2} \leqslant \\ &\leqslant kmT+\sqrt{\gamma kmTp}\, \left[ \sum_{a=0}^{p-1}|S_1|^2 \sum_{a=0}^{p-1}|S_2|^2 \right]^{1/4} < pkm. \end{aligned} \]
Thus there exists an \(a\) for which
\[ |S_1|\geqslant \gamma_1 k,\qquad |S_2|\geqslant \gamma_2 m, \tag{5} \]
where \(\gamma_1\gamma_2=\gamma\).
From (5) and the lemma there follows the existence of integers \(u,u_1\), and \(v\) such that, with the numbers (2), the numbers
\[ u_1+vt,\qquad 0\leqslant t\leqslant \frac{p-3}{2} \tag{6} \]
are congruent respectively to \(k_1\) numbers from \(K\) and \(m_1\) numbers from \(M\), where
\[ k_1\geqslant \frac{1+\gamma_1}{2}\,k,\qquad m_1\geqslant \frac{1+\gamma_2}{2}\,m. \]
Let the numbers from (6) congruent to numbers from \(M\) be obtained for the values \(t\) equal to \(t_0,t_1,\ldots,t_{m_1-1}\), with \(t_j<t_{j+1}\), \(j=0,1,\ldots,m_1-2\), \(t_0=0\).
If \(\max(s_{k_1-1},t_{m_1-1})\geqslant k_1+m_1-3\), then, in view of Theorem 4,
\[ T_p(K+M)\geqslant k_1+m_1+\min(k_1,m_1)-3. \]
In view of (4), this contradicts (3).
If \(\max(s_{k_1-1},t_{m_1-1})\leqslant k_1+m_1-4\), then every number from \(K\) (respectively \(M\)) is congruent to one of the numbers \(u+vs\) (respectively \(u_1+vt\)) for
\[ -(k_1+m_1-4)\leqslant s,\quad t\leqslant 2k_1+2m_1-8. \]
Applying Theorem 4, we complete the proof.
Steklov Mathematical Institute
of the Academy of Sciences of the USSR
Received
19 VI 1961
REFERENCES
- A. G. Vosper, J. London Math. Soc., 31, No. 122, 200 (1956).
- G. A. Freiman, Izv. Vyssh. uchebn. zaved., Matem., No. 6 (13), 202 (1959).
- G. A. Freiman, Uch. zap. Elabuzhsk. gos. ped. inst., 8, 72 (1960).